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Hypothesis Testing: Uses, Steps & Example

By Jim Frost 4 Comments

What is Hypothesis Testing?

Hypothesis testing in statistics uses sample data to infer the properties of a whole population . These tests determine whether a random sample provides sufficient evidence to conclude an effect or relationship exists in the population. Researchers use them to help separate genuine population-level effects from false effects that random chance can create in samples. These methods are also known as significance testing.

For example, researchers are testing a new medication to see if it lowers blood pressure. They compare a group taking the drug to a control group taking a placebo. If their hypothesis test results are statistically significant, the medication’s effect of lowering blood pressure likely exists in the broader population, not just the sample studied.

Using Hypothesis Tests

A hypothesis test evaluates two mutually exclusive statements about a population to determine which statement the sample data best supports. These two statements are called the null hypothesis and the alternative hypothesis . The following are typical examples:

• Null Hypothesis : The effect does not exist in the population.
• Alternative Hypothesis : The effect does exist in the population.

Hypothesis testing accounts for the inherent uncertainty of using a sample to draw conclusions about a population, which reduces the chances of false discoveries. These procedures determine whether the sample data are sufficiently inconsistent with the null hypothesis that you can reject it. If you can reject the null, your data favor the alternative statement that an effect exists in the population.

Statistical significance in hypothesis testing indicates that an effect you see in sample data also likely exists in the population after accounting for random sampling error , variability, and sample size. Your results are statistically significant when the p-value is less than your significance level or, equivalently, when your confidence interval excludes the null hypothesis value.

Conversely, non-significant results indicate that despite an apparent sample effect, you can’t be sure it exists in the population. It could be chance variation in the sample and not a genuine effect.

Learn more about Failing to Reject the Null .

5 Steps of Significance Testing

Hypothesis testing involves five key steps, each critical to validating a research hypothesis using statistical methods:

• Formulate the Hypotheses : Write your research hypotheses as a null hypothesis (H 0 ) and an alternative hypothesis (H A ).
• Data Collection : Gather data specifically aimed at testing the hypothesis.
• Conduct A Test : Use a suitable statistical test to analyze your data.
• Make a Decision : Based on the statistical test results, decide whether to reject the null hypothesis or fail to reject it.
• Report the Results : Summarize and present the outcomes in your report’s results and discussion sections.

While the specifics of these steps can vary depending on the research context and the data type, the fundamental process of hypothesis testing remains consistent across different studies.

Let’s work through these steps in an example!

Hypothesis Testing Example

Researchers want to determine if a new educational program improves student performance on standardized tests. They randomly assign 30 students to a control group , which follows the standard curriculum, and another 30 students to a treatment group, which participates in the new educational program. After a semester, they compare the test scores of both groups.

Download the CSV data file to perform the hypothesis testing yourself: Hypothesis_Testing .

The researchers write their hypotheses. These statements apply to the population, so they use the mu (μ) symbol for the population mean parameter .

• Null Hypothesis (H 0 ) : The population means of the test scores for the two groups are equal (μ 1 = μ 2 ).
• Alternative Hypothesis (H A ) : The population means of the test scores for the two groups are unequal (μ 1 ≠ μ 2 ).

Choosing the correct hypothesis test depends on attributes such as data type and number of groups. Because they’re using continuous data and comparing two means, the researchers use a 2-sample t-test .

Here are the results.

The treatment group’s mean is 58.70, compared to the control group’s mean of 48.12. The mean difference is 10.67 points. Use the test’s p-value and significance level to determine whether this difference is likely a product of random fluctuation in the sample or a genuine population effect.

Because the p-value (0.000) is less than the standard significance level of 0.05, the results are statistically significant, and we can reject the null hypothesis. The sample data provides sufficient evidence to conclude that the new program’s effect exists in the population.

Limitations

Hypothesis testing improves your effectiveness in making data-driven decisions. However, it is not 100% accurate because random samples occasionally produce fluky results. Hypothesis tests have two types of errors, both relating to drawing incorrect conclusions.

• Type I error: The test rejects a true null hypothesis—a false positive.
• Type II error: The test fails to reject a false null hypothesis—a false negative.

Learn more about Type I and Type II Errors .

Our exploration of hypothesis testing using a practical example of an educational program reveals its powerful ability to guide decisions based on statistical evidence. Whether you’re a student, researcher, or professional, understanding and applying these procedures can open new doors to discovering insights and making informed decisions. Let this tool empower your analytical endeavors as you navigate through the vast seas of data.

Learn more about the Hypothesis Tests for Various Data Types .

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Reader Interactions

June 10, 2024 at 10:51 am

Thank you, Jim, for another helpful article; timely too since I have started reading your new book on hypothesis testing and, now that we are at the end of the school year, my district is asking me to perform a number of evaluations on instructional programs. This is where my question/concern comes in. You mention that hypothesis testing is all about testing samples. However, I use all the students in my district when I make these comparisons. Since I am using the entire “population” in my evaluations (I don’t select a sample of third grade students, for example, but I use all 700 third graders), am I somehow misusing the tests? Or can I rest assured that my district’s student population is only a sample of the universal population of students?

June 10, 2024 at 1:50 pm

I hope you are finding the book helpful!

Yes, the purpose of hypothesis testing is to infer the properties of a population while accounting for random sampling error.

In your case, it comes down to how you want to use the results. Who do you want the results to apply to?

If you’re summarizing the sample, looking for trends and patterns, or evaluating those students and don’t plan to apply those results to other students, you don’t need hypothesis testing because there is no sampling error. They are the population and you can just use descriptive statistics. In this case, you’d only need to focus on the practical significance of the effect sizes.

On the other hand, if you want to apply the results from this group to other students, you’ll need hypothesis testing. However, there is the complicating issue of what population your sample of students represent. I’m sure your district has its own unique characteristics, demographics, etc. Your district’s students probably don’t adequately represent a universal population. At the very least, you’d need to recognize any special attributes of your district and how they could bias the results when trying to apply them outside the district. Or they might apply to similar districts in your region.

However, I’d imagine your 3rd graders probably adequately represent future classes of 3rd graders in your district. You need to be alert to changing demographics. At least in the short run I’d imagine they’d be representative of future classes.

Think about how these results will be used. Do they just apply to the students you measured? Then you don’t need hypothesis tests. However, if the results are being used to infer things about other students outside of the sample, you’ll need hypothesis testing along with considering how well your students represent the other students and how they differ.

I hope that helps!

June 10, 2024 at 3:21 pm

Thank you so much, Jim, for the suggestions in terms of what I need to think about and consider! You are always so clear in your explanations!!!!

June 10, 2024 at 3:22 pm

You’re very welcome! Best of luck with your evaluations!

Comments and Questions Cancel reply

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Understanding Hypothesis Tests: Why We Need to Use Hypothesis Tests in Statistics

Topics: Hypothesis Testing , Data Analysis , Statistics

Hypothesis testing is an essential procedure in statistics. A hypothesis test evaluates two mutually exclusive statements about a population to determine which statement is best supported by the sample data. When we say that a finding is statistically significant, it’s thanks to a hypothesis test. How do these tests really work and what does statistical significance actually mean?

In this series of three posts, I’ll help you intuitively understand how hypothesis tests work by focusing on concepts and graphs rather than equations and numbers. After all, a key reason to use statistical software like Minitab is so you don’t get bogged down in the calculations and can instead focus on understanding your results.

To kick things off in this post, I highlight the rationale for using hypothesis tests with an example.

The Scenario

An economist wants to determine whether the monthly energy cost for families has changed from the previous year, when the mean cost per month was $260. The economist randomly samples 25 families and records their energy costs for the current year. (The data for this example is FamilyEnergyCost and it is just one of the many data set examples that can be found in Minitab’s Data Set Library.)

I’ll use these descriptive statistics to create a probability distribution plot that shows you the importance of hypothesis tests. Read on!

The Need for Hypothesis Tests

Why do we even need hypothesis tests? After all, we took a random sample and our sample mean of 330.6 is different from 260. That is different, right? Unfortunately, the picture is muddied because we’re looking at a sample rather than the entire population.

Sampling error is the difference between a sample and the entire population. Thanks to sampling error, it’s entirely possible that while our sample mean is 330.6, the population mean could still be 260. Or, to put it another way, if we repeated the experiment, it’s possible that the second sample mean could be close to 260. A hypothesis test helps assess the likelihood of this possibility!

Use the Sampling Distribution to See If Our Sample Mean is Unlikely

For any given random sample, the mean of the sample almost certainly doesn’t equal the true mean of the population due to sampling error. For our example, it’s unlikely that the mean cost for the entire population is exactly 330.6. In fact, if we took multiple random samples of the same size from the same population, we could plot a distribution of the sample means.

A sampling distribution is the distribution of a statistic, such as the mean, that is obtained by repeatedly drawing a large number of samples from a specific population. This distribution allows you to determine the probability of obtaining the sample statistic.

Fortunately, I can create a plot of sample means without collecting many different random samples! Instead, I’ll create a probability distribution plot using the t-distribution , the sample size, and the variability in our sample to graph the sampling distribution.

Our goal is to determine whether our sample mean is significantly different from the null hypothesis mean. Therefore, we’ll use the graph to see whether our sample mean of 330.6 is unlikely assuming that the population mean is 260. The graph below shows the expected distribution of sample means.

You can see that the most probable sample mean is 260, which makes sense because we’re assuming that the null hypothesis is true. However, there is a reasonable probability of obtaining a sample mean that ranges from 167 to 352, and even beyond! The takeaway from this graph is that while our sample mean of 330.6 is not the most probable, it’s also not outside the realm of possibility.

The Role of Hypothesis Tests

We’ve placed our sample mean in the context of all possible sample means while assuming that the null hypothesis is true. Are these results statistically significant?

As you can see, there is no magic place on the distribution curve to make this determination. Instead, we have a continual decrease in the probability of obtaining sample means that are further from the null hypothesis value. Where do we draw the line?

This is where hypothesis tests are useful. A hypothesis test allows us quantify the probability that our sample mean is unusual.

For this series of posts, I’ll continue to use this graphical framework and add in the significance level, P value, and confidence interval to show how hypothesis tests work and what statistical significance really means.

• Part Two: Significance Levels (alpha) and P values
• Part Three: Confidence Intervals and Confidence Levels

If you'd like to see how I made these graphs, please read: How to Create a Graphical Version of the 1-sample t-Test .

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Hypothesis Testing – A Deep Dive into Hypothesis Testing, The Backbone of Statistical Inference

• September 21, 2023

Explore the intricacies of hypothesis testing, a cornerstone of statistical analysis. Dive into methods, interpretations, and applications for making data-driven decisions.

In this Blog post we will learn:

• What is Hypothesis Testing?
• Steps in Hypothesis Testing 2.1. Set up Hypotheses: Null and Alternative 2.2. Choose a Significance Level (α) 2.3. Calculate a test statistic and P-Value 2.4. Make a Decision
• Example : Testing a new drug.
• Example in python

1. What is Hypothesis Testing?

In simple terms, hypothesis testing is a method used to make decisions or inferences about population parameters based on sample data. Imagine being handed a dice and asked if it’s biased. By rolling it a few times and analyzing the outcomes, you’d be engaging in the essence of hypothesis testing.

Think of hypothesis testing as the scientific method of the statistics world. Suppose you hear claims like “This new drug works wonders!” or “Our new website design boosts sales.” How do you know if these statements hold water? Enter hypothesis testing.

2. Steps in Hypothesis Testing

• Set up Hypotheses : Begin with a null hypothesis (H0) and an alternative hypothesis (Ha).
• Choose a Significance Level (α) : Typically 0.05, this is the probability of rejecting the null hypothesis when it’s actually true. Think of it as the chance of accusing an innocent person.
• Calculate Test statistic and P-Value : Gather evidence (data) and calculate a test statistic.
• p-value : This is the probability of observing the data, given that the null hypothesis is true. A small p-value (typically ≤ 0.05) suggests the data is inconsistent with the null hypothesis.
• Decision Rule : If the p-value is less than or equal to α, you reject the null hypothesis in favor of the alternative.

2.1. Set up Hypotheses: Null and Alternative

Before diving into testing, we must formulate hypotheses. The null hypothesis (H0) represents the default assumption, while the alternative hypothesis (H1) challenges it.

For instance, in drug testing, H0 : “The new drug is no better than the existing one,” H1 : “The new drug is superior .”

2.2. Choose a Significance Level (α)

When You collect and analyze data to test H0 and H1 hypotheses. Based on your analysis, you decide whether to reject the null hypothesis in favor of the alternative, or fail to reject / Accept the null hypothesis.

The significance level, often denoted by $α$, represents the probability of rejecting the null hypothesis when it is actually true.

In other words, it’s the risk you’re willing to take of making a Type I error (false positive).

Type I Error (False Positive) :

• Symbolized by the Greek letter alpha (α).
• Occurs when you incorrectly reject a true null hypothesis . In other words, you conclude that there is an effect or difference when, in reality, there isn’t.
• The probability of making a Type I error is denoted by the significance level of a test. Commonly, tests are conducted at the 0.05 significance level , which means there’s a 5% chance of making a Type I error .
• Commonly used significance levels are 0.01, 0.05, and 0.10, but the choice depends on the context of the study and the level of risk one is willing to accept.

Example : If a drug is not effective (truth), but a clinical trial incorrectly concludes that it is effective (based on the sample data), then a Type I error has occurred.

Type II Error (False Negative) :

• Symbolized by the Greek letter beta (β).
• Occurs when you accept a false null hypothesis . This means you conclude there is no effect or difference when, in reality, there is.
• The probability of making a Type II error is denoted by β. The power of a test (1 – β) represents the probability of correctly rejecting a false null hypothesis.

Example : If a drug is effective (truth), but a clinical trial incorrectly concludes that it is not effective (based on the sample data), then a Type II error has occurred.

Balancing the Errors :

In practice, there’s a trade-off between Type I and Type II errors. Reducing the risk of one typically increases the risk of the other. For example, if you want to decrease the probability of a Type I error (by setting a lower significance level), you might increase the probability of a Type II error unless you compensate by collecting more data or making other adjustments.

It’s essential to understand the consequences of both types of errors in any given context. In some situations, a Type I error might be more severe, while in others, a Type II error might be of greater concern. This understanding guides researchers in designing their experiments and choosing appropriate significance levels.

2.3. Calculate a test statistic and P-Value

Test statistic : A test statistic is a single number that helps us understand how far our sample data is from what we’d expect under a null hypothesis (a basic assumption we’re trying to test against). Generally, the larger the test statistic, the more evidence we have against our null hypothesis. It helps us decide whether the differences we observe in our data are due to random chance or if there’s an actual effect.

P-value : The P-value tells us how likely we would get our observed results (or something more extreme) if the null hypothesis were true. It’s a value between 0 and 1. – A smaller P-value (typically below 0.05) means that the observation is rare under the null hypothesis, so we might reject the null hypothesis. – A larger P-value suggests that what we observed could easily happen by random chance, so we might not reject the null hypothesis.

2.4. Make a Decision

Relationship between $α$ and P-Value

When conducting a hypothesis test:

We then calculate the p-value from our sample data and the test statistic.

Finally, we compare the p-value to our chosen $α$:

• If $p−value≤α$: We reject the null hypothesis in favor of the alternative hypothesis. The result is said to be statistically significant.
• If $p−value>α$: We fail to reject the null hypothesis. There isn’t enough statistical evidence to support the alternative hypothesis.

3. Example : Testing a new drug.

Imagine we are investigating whether a new drug is effective at treating headaches faster than drug B.

Setting Up the Experiment : You gather 100 people who suffer from headaches. Half of them (50 people) are given the new drug (let’s call this the ‘Drug Group’), and the other half are given a sugar pill, which doesn’t contain any medication.

• Set up Hypotheses : Before starting, you make a prediction:
• Null Hypothesis (H0): The new drug has no effect. Any difference in healing time between the two groups is just due to random chance.
• Alternative Hypothesis (H1): The new drug does have an effect. The difference in healing time between the two groups is significant and not just by chance.

Calculate Test statistic and P-Value : After the experiment, you analyze the data. The “test statistic” is a number that helps you understand the difference between the two groups in terms of standard units.

For instance, let’s say:

• The average healing time in the Drug Group is 2 hours.
• The average healing time in the Placebo Group is 3 hours.

The test statistic helps you understand how significant this 1-hour difference is. If the groups are large and the spread of healing times in each group is small, then this difference might be significant. But if there’s a huge variation in healing times, the 1-hour difference might not be so special.

Imagine the P-value as answering this question: “If the new drug had NO real effect, what’s the probability that I’d see a difference as extreme (or more extreme) as the one I found, just by random chance?”

For instance:

• P-value of 0.01 means there’s a 1% chance that the observed difference (or a more extreme difference) would occur if the drug had no effect. That’s pretty rare, so we might consider the drug effective.
• P-value of 0.5 means there’s a 50% chance you’d see this difference just by chance. That’s pretty high, so we might not be convinced the drug is doing much.
• If the P-value is less than ($α$) 0.05: the results are “statistically significant,” and they might reject the null hypothesis , believing the new drug has an effect.
• If the P-value is greater than ($α$) 0.05: the results are not statistically significant, and they don’t reject the null hypothesis , remaining unsure if the drug has a genuine effect.

4. Example in python

For simplicity, let’s say we’re using a t-test (common for comparing means). Let’s dive into Python:

Making a Decision : “The results are statistically significant! p-value < 0.05 , The drug seems to have an effect!” If not, we’d say, “Looks like the drug isn’t as miraculous as we thought.”

5. Conclusion

Hypothesis testing is an indispensable tool in data science, allowing us to make data-driven decisions with confidence. By understanding its principles, conducting tests properly, and considering real-world applications, you can harness the power of hypothesis testing to unlock valuable insights from your data.

More Articles

Correlation – connecting the dots, the role of correlation in data analysis, sampling and sampling distributions – a comprehensive guide on sampling and sampling distributions, law of large numbers – a deep dive into the world of statistics, central limit theorem – a deep dive into central limit theorem and its significance in statistics, skewness and kurtosis – peaks and tails, understanding data through skewness and kurtosis”, similar articles, complete introduction to linear regression in r, how to implement common statistical significance tests and find the p value, logistic regression – a complete tutorial with examples in r.

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6a.2 - steps for hypothesis tests, the logic of hypothesis testing section  .

A hypothesis, in statistics, is a statement about a population parameter, where this statement typically is represented by some specific numerical value. In testing a hypothesis, we use a method where we gather data in an effort to gather evidence about the hypothesis.

How do we decide whether to reject the null hypothesis?

• If the sample data are consistent with the null hypothesis, then we do not reject it.
• If the sample data are inconsistent with the null hypothesis, but consistent with the alternative, then we reject the null hypothesis and conclude that the alternative hypothesis is true.

Six Steps for Hypothesis Tests Section  

In hypothesis testing, there are certain steps one must follow. Below these are summarized into six such steps to conducting a test of a hypothesis.

• Set up the hypotheses and check conditions : Each hypothesis test includes two hypotheses about the population. One is the null hypothesis, notated as \(H_0 \), which is a statement of a particular parameter value. This hypothesis is assumed to be true until there is evidence to suggest otherwise. The second hypothesis is called the alternative, or research hypothesis, notated as \(H_a \). The alternative hypothesis is a statement of a range of alternative values in which the parameter may fall. One must also check that any conditions (assumptions) needed to run the test have been satisfied e.g. normality of data, independence, and number of success and failure outcomes.
• Decide on the significance level, \(\alpha \): This value is used as a probability cutoff for making decisions about the null hypothesis. This alpha value represents the probability we are willing to place on our test for making an incorrect decision in regards to rejecting the null hypothesis. The most common \(\alpha \) value is 0.05 or 5%. Other popular choices are 0.01 (1%) and 0.1 (10%).
• Calculate the test statistic: Gather sample data and calculate a test statistic where the sample statistic is compared to the parameter value. The test statistic is calculated under the assumption the null hypothesis is true and incorporates a measure of standard error and assumptions (conditions) related to the sampling distribution.
• Calculate probability value (p-value), or find the rejection region: A p-value is found by using the test statistic to calculate the probability of the sample data producing such a test statistic or one more extreme. The rejection region is found by using alpha to find a critical value; the rejection region is the area that is more extreme than the critical value. We discuss the p-value and rejection region in more detail in the next section.
• Make a decision about the null hypothesis: In this step, we decide to either reject the null hypothesis or decide to fail to reject the null hypothesis. Notice we do not make a decision where we will accept the null hypothesis.
• State an overall conclusion : Once we have found the p-value or rejection region, and made a statistical decision about the null hypothesis (i.e. we will reject the null or fail to reject the null), we then want to summarize our results into an overall conclusion for our test.

We will follow these six steps for the remainder of this Lesson. In the future Lessons, the steps will be followed but may not be explained explicitly.

Step 1 is a very important step to set up correctly. If your hypotheses are incorrect, your conclusion will be incorrect. In this next section, we practice with Step 1 for the one sample situations.

• Hypothesis Testing: Definition, Uses, Limitations + Examples

Hypothesis testing is as old as the scientific method and is at the heart of the research process. 

Research exists to validate or disprove assumptions about various phenomena. The process of validation involves testing and it is in this context that we will explore hypothesis testing. 

What is a Hypothesis? 

A hypothesis is a calculated prediction or assumption about a population parameter based on limited evidence. The whole idea behind hypothesis formulation is testing—this means the researcher subjects his or her calculated assumption to a series of evaluations to know whether they are true or false. 

Typically, every research starts with a hypothesis—the investigator makes a claim and experiments to prove that this claim is true or false . For instance, if you predict that students who drink milk before class perform better than those who don’t, then this becomes a hypothesis that can be confirmed or refuted using an experiment.  

Read: What is Empirical Research Study? [Examples & Method]

What are the Types of Hypotheses? 

1. simple hypothesis.

Also known as a basic hypothesis, a simple hypothesis suggests that an independent variable is responsible for a corresponding dependent variable. In other words, an occurrence of the independent variable inevitably leads to an occurrence of the dependent variable. 

Typically, simple hypotheses are considered as generally true, and they establish a causal relationship between two variables. 

Examples of Simple Hypothesis  

• Drinking soda and other sugary drinks can cause obesity. 
• Smoking cigarettes daily leads to lung cancer.

2. Complex Hypothesis

A complex hypothesis is also known as a modal. It accounts for the causal relationship between two independent variables and the resulting dependent variables. This means that the combination of the independent variables leads to the occurrence of the dependent variables . 

Examples of Complex Hypotheses  

• Adults who do not smoke and drink are less likely to develop liver-related conditions.
• Global warming causes icebergs to melt which in turn causes major changes in weather patterns.

3. Null Hypothesis

As the name suggests, a null hypothesis is formed when a researcher suspects that there’s no relationship between the variables in an observation. In this case, the purpose of the research is to approve or disapprove this assumption. 

Examples of Null Hypothesis

• This is no significant change in a student’s performance if they drink coffee or tea before classes. 
• There’s no significant change in the growth of a plant if one uses distilled water only or vitamin-rich water. 

Read: Research Report: Definition, Types + [Writing Guide]

4. Alternative Hypothesis 

To disapprove a null hypothesis, the researcher has to come up with an opposite assumption—this assumption is known as the alternative hypothesis. This means if the null hypothesis says that A is false, the alternative hypothesis assumes that A is true. 

An alternative hypothesis can be directional or non-directional depending on the direction of the difference. A directional alternative hypothesis specifies the direction of the tested relationship, stating that one variable is predicted to be larger or smaller than the null value while a non-directional hypothesis only validates the existence of a difference without stating its direction. 

Examples of Alternative Hypotheses  

• Starting your day with a cup of tea instead of a cup of coffee can make you more alert in the morning. 
• The growth of a plant improves significantly when it receives distilled water instead of vitamin-rich water. 

5. Logical Hypothesis

Logical hypotheses are some of the most common types of calculated assumptions in systematic investigations. It is an attempt to use your reasoning to connect different pieces in research and build a theory using little evidence. In this case, the researcher uses any data available to him, to form a plausible assumption that can be tested. 

Examples of Logical Hypothesis

• Waking up early helps you to have a more productive day. 
• Beings from Mars would not be able to breathe the air in the atmosphere of the Earth. 

6. Empirical Hypothesis  

After forming a logical hypothesis, the next step is to create an empirical or working hypothesis. At this stage, your logical hypothesis undergoes systematic testing to prove or disprove the assumption. An empirical hypothesis is subject to several variables that can trigger changes and lead to specific outcomes. 

Examples of Empirical Testing 

• People who eat more fish run faster than people who eat meat.
• Women taking vitamin E grow hair faster than those taking vitamin K.

7. Statistical Hypothesis

When forming a statistical hypothesis, the researcher examines the portion of a population of interest and makes a calculated assumption based on the data from this sample. A statistical hypothesis is most common with systematic investigations involving a large target audience. Here, it’s impossible to collect responses from every member of the population so you have to depend on data from your sample and extrapolate the results to the wider population. 

Examples of Statistical Hypothesis  

• 45% of students in Louisiana have middle-income parents. 
• 80% of the UK’s population gets a divorce because of irreconcilable differences.

What is Hypothesis Testing? 

Hypothesis testing is an assessment method that allows researchers to determine the plausibility of a hypothesis. It involves testing an assumption about a specific population parameter to know whether it’s true or false. These population parameters include variance, standard deviation, and median. 

Typically, hypothesis testing starts with developing a null hypothesis and then performing several tests that support or reject the null hypothesis. The researcher uses test statistics to compare the association or relationship between two or more variables. 

Explore: Research Bias: Definition, Types + Examples

Researchers also use hypothesis testing to calculate the coefficient of variation and determine if the regression relationship and the correlation coefficient are statistically significant.

How Hypothesis Testing Works

The basis of hypothesis testing is to examine and analyze the null hypothesis and alternative hypothesis to know which one is the most plausible assumption. Since both assumptions are mutually exclusive, only one can be true. In other words, the occurrence of a null hypothesis destroys the chances of the alternative coming to life, and vice-versa. 

Interesting: 21 Chrome Extensions for Academic Researchers in 2021

What Are The Stages of Hypothesis Testing?  

To successfully confirm or refute an assumption, the researcher goes through five (5) stages of hypothesis testing; 

• Determine the null hypothesis
• Specify the alternative hypothesis
• Set the significance level
• Calculate the test statistics and corresponding P-value
• Draw your conclusion
• Determine the Null Hypothesis

Like we mentioned earlier, hypothesis testing starts with creating a null hypothesis which stands as an assumption that a certain statement is false or implausible. For example, the null hypothesis (H0) could suggest that different subgroups in the research population react to a variable in the same way. 

• Specify the Alternative Hypothesis

Once you know the variables for the null hypothesis, the next step is to determine the alternative hypothesis. The alternative hypothesis counters the null assumption by suggesting the statement or assertion is true. Depending on the purpose of your research, the alternative hypothesis can be one-sided or two-sided. 

Using the example we established earlier, the alternative hypothesis may argue that the different sub-groups react differently to the same variable based on several internal and external factors. 

• Set the Significance Level

Many researchers create a 5% allowance for accepting the value of an alternative hypothesis, even if the value is untrue. This means that there is a 0.05 chance that one would go with the value of the alternative hypothesis, despite the truth of the null hypothesis. 

Something to note here is that the smaller the significance level, the greater the burden of proof needed to reject the null hypothesis and support the alternative hypothesis.

Explore: What is Data Interpretation? + [Types, Method & Tools]

• Calculate the Test Statistics and Corresponding P-Value 

Test statistics in hypothesis testing allow you to compare different groups between variables while the p-value accounts for the probability of obtaining sample statistics if your null hypothesis is true. In this case, your test statistics can be the mean, median and similar parameters. 

If your p-value is 0.65, for example, then it means that the variable in your hypothesis will happen 65 in100 times by pure chance. Use this formula to determine the p-value for your data: 

• Draw Your Conclusions

After conducting a series of tests, you should be able to agree or refute the hypothesis based on feedback and insights from your sample data.  

Applications of Hypothesis Testing in Research

Hypothesis testing isn’t only confined to numbers and calculations; it also has several real-life applications in business, manufacturing, advertising, and medicine. 

In a factory or other manufacturing plants, hypothesis testing is an important part of quality and production control before the final products are approved and sent out to the consumer. 

During ideation and strategy development, C-level executives use hypothesis testing to evaluate their theories and assumptions before any form of implementation. For example, they could leverage hypothesis testing to determine whether or not some new advertising campaign, marketing technique, etc. causes increased sales. 

In addition, hypothesis testing is used during clinical trials to prove the efficacy of a drug or new medical method before its approval for widespread human usage. 

What is an Example of Hypothesis Testing?

An employer claims that her workers are of above-average intelligence. She takes a random sample of 20 of them and gets the following results: 

Mean IQ Scores: 110

Standard Deviation: 15 

Mean Population IQ: 100

Step 1: Using the value of the mean population IQ, we establish the null hypothesis as 100.

Step 2: State that the alternative hypothesis is greater than 100.

Step 3: State the alpha level as 0.05 or 5% 

Step 4: Find the rejection region area (given by your alpha level above) from the z-table. An area of .05 is equal to a z-score of 1.645.

Step 5: Calculate the test statistics using this formula

Z = (110–100) ÷ (15÷√20) 

10 ÷ 3.35 = 2.99 

If the value of the test statistics is higher than the value of the rejection region, then you should reject the null hypothesis. If it is less, then you cannot reject the null. 

In this case, 2.99 > 1.645 so we reject the null. 

Importance/Benefits of Hypothesis Testing 

The most significant benefit of hypothesis testing is it allows you to evaluate the strength of your claim or assumption before implementing it in your data set. Also, hypothesis testing is the only valid method to prove that something “is or is not”. Other benefits include: 

• Hypothesis testing provides a reliable framework for making any data decisions for your population of interest. 
• It helps the researcher to successfully extrapolate data from the sample to the larger population. 
• Hypothesis testing allows the researcher to determine whether the data from the sample is statistically significant. 
• Hypothesis testing is one of the most important processes for measuring the validity and reliability of outcomes in any systematic investigation. 
• It helps to provide links to the underlying theory and specific research questions.

Criticism and Limitations of Hypothesis Testing

Several limitations of hypothesis testing can affect the quality of data you get from this process. Some of these limitations include: 

• The interpretation of a p-value for observation depends on the stopping rule and definition of multiple comparisons. This makes it difficult to calculate since the stopping rule is subject to numerous interpretations, plus “multiple comparisons” are unavoidably ambiguous. 
• Conceptual issues often arise in hypothesis testing, especially if the researcher merges Fisher and Neyman-Pearson’s methods which are conceptually distinct. 
• In an attempt to focus on the statistical significance of the data, the researcher might ignore the estimation and confirmation by repeated experiments.
• Hypothesis testing can trigger publication bias, especially when it requires statistical significance as a criterion for publication.
• When used to detect whether a difference exists between groups, hypothesis testing can trigger absurd assumptions that affect the reliability of your observation.

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Hypothesis Testing

Hypothesis testing is a tool for making statistical inferences about the population data. It is an analysis tool that tests assumptions and determines how likely something is within a given standard of accuracy. Hypothesis testing provides a way to verify whether the results of an experiment are valid.

A null hypothesis and an alternative hypothesis are set up before performing the hypothesis testing. This helps to arrive at a conclusion regarding the sample obtained from the population. In this article, we will learn more about hypothesis testing, its types, steps to perform the testing, and associated examples.

What is Hypothesis Testing in Statistics?

Hypothesis testing uses sample data from the population to draw useful conclusions regarding the population probability distribution . It tests an assumption made about the data using different types of hypothesis testing methodologies. The hypothesis testing results in either rejecting or not rejecting the null hypothesis.

Hypothesis Testing Definition

Hypothesis testing can be defined as a statistical tool that is used to identify if the results of an experiment are meaningful or not. It involves setting up a null hypothesis and an alternative hypothesis. These two hypotheses will always be mutually exclusive. This means that if the null hypothesis is true then the alternative hypothesis is false and vice versa. An example of hypothesis testing is setting up a test to check if a new medicine works on a disease in a more efficient manner.

Null Hypothesis

The null hypothesis is a concise mathematical statement that is used to indicate that there is no difference between two possibilities. In other words, there is no difference between certain characteristics of data. This hypothesis assumes that the outcomes of an experiment are based on chance alone. It is denoted as \(H_{0}\). Hypothesis testing is used to conclude if the null hypothesis can be rejected or not. Suppose an experiment is conducted to check if girls are shorter than boys at the age of 5. The null hypothesis will say that they are the same height.

Alternative Hypothesis

The alternative hypothesis is an alternative to the null hypothesis. It is used to show that the observations of an experiment are due to some real effect. It indicates that there is a statistical significance between two possible outcomes and can be denoted as \(H_{1}\) or \(H_{a}\). For the above-mentioned example, the alternative hypothesis would be that girls are shorter than boys at the age of 5.

Hypothesis Testing P Value

In hypothesis testing, the p value is used to indicate whether the results obtained after conducting a test are statistically significant or not. It also indicates the probability of making an error in rejecting or not rejecting the null hypothesis.This value is always a number between 0 and 1. The p value is compared to an alpha level, \(\alpha\) or significance level. The alpha level can be defined as the acceptable risk of incorrectly rejecting the null hypothesis. The alpha level is usually chosen between 1% to 5%.

Hypothesis Testing Critical region

All sets of values that lead to rejecting the null hypothesis lie in the critical region. Furthermore, the value that separates the critical region from the non-critical region is known as the critical value.

Hypothesis Testing Formula

Depending upon the type of data available and the size, different types of hypothesis testing are used to determine whether the null hypothesis can be rejected or not. The hypothesis testing formula for some important test statistics are given below:

• z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\). \(\overline{x}\) is the sample mean, \(\mu\) is the population mean, \(\sigma\) is the population standard deviation and n is the size of the sample.
• t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\). s is the sample standard deviation.
• \(\chi ^{2} = \sum \frac{(O_{i}-E_{i})^{2}}{E_{i}}\). \(O_{i}\) is the observed value and \(E_{i}\) is the expected value.

We will learn more about these test statistics in the upcoming section.

Types of Hypothesis Testing

Selecting the correct test for performing hypothesis testing can be confusing. These tests are used to determine a test statistic on the basis of which the null hypothesis can either be rejected or not rejected. Some of the important tests used for hypothesis testing are given below.

Hypothesis Testing Z Test

A z test is a way of hypothesis testing that is used for a large sample size (n ≥ 30). It is used to determine whether there is a difference between the population mean and the sample mean when the population standard deviation is known. It can also be used to compare the mean of two samples. It is used to compute the z test statistic. The formulas are given as follows:

• One sample: z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\).
• Two samples: z = \(\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}\).

Hypothesis Testing t Test

The t test is another method of hypothesis testing that is used for a small sample size (n < 30). It is also used to compare the sample mean and population mean. However, the population standard deviation is not known. Instead, the sample standard deviation is known. The mean of two samples can also be compared using the t test.

• One sample: t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\).
• Two samples: t = \(\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{s_{1}^{2}}{n_{1}}+\frac{s_{2}^{2}}{n_{2}}}}\).

Hypothesis Testing Chi Square

The Chi square test is a hypothesis testing method that is used to check whether the variables in a population are independent or not. It is used when the test statistic is chi-squared distributed.

One Tailed Hypothesis Testing

One tailed hypothesis testing is done when the rejection region is only in one direction. It can also be known as directional hypothesis testing because the effects can be tested in one direction only. This type of testing is further classified into the right tailed test and left tailed test.

Right Tailed Hypothesis Testing

The right tail test is also known as the upper tail test. This test is used to check whether the population parameter is greater than some value. The null and alternative hypotheses for this test are given as follows:

\(H_{0}\): The population parameter is ≤ some value

\(H_{1}\): The population parameter is > some value.

If the test statistic has a greater value than the critical value then the null hypothesis is rejected

Left Tailed Hypothesis Testing

The left tail test is also known as the lower tail test. It is used to check whether the population parameter is less than some value. The hypotheses for this hypothesis testing can be written as follows:

\(H_{0}\): The population parameter is ≥ some value

\(H_{1}\): The population parameter is < some value.

The null hypothesis is rejected if the test statistic has a value lesser than the critical value.

Two Tailed Hypothesis Testing

In this hypothesis testing method, the critical region lies on both sides of the sampling distribution. It is also known as a non - directional hypothesis testing method. The two-tailed test is used when it needs to be determined if the population parameter is assumed to be different than some value. The hypotheses can be set up as follows:

\(H_{0}\): the population parameter = some value

\(H_{1}\): the population parameter ≠ some value

The null hypothesis is rejected if the test statistic has a value that is not equal to the critical value.

Hypothesis Testing Steps

Hypothesis testing can be easily performed in five simple steps. The most important step is to correctly set up the hypotheses and identify the right method for hypothesis testing. The basic steps to perform hypothesis testing are as follows:

• Step 1: Set up the null hypothesis by correctly identifying whether it is the left-tailed, right-tailed, or two-tailed hypothesis testing.
• Step 2: Set up the alternative hypothesis.
• Step 3: Choose the correct significance level, \(\alpha\), and find the critical value.
• Step 4: Calculate the correct test statistic (z, t or \(\chi\)) and p-value.
• Step 5: Compare the test statistic with the critical value or compare the p-value with \(\alpha\) to arrive at a conclusion. In other words, decide if the null hypothesis is to be rejected or not.

Hypothesis Testing Example

The best way to solve a problem on hypothesis testing is by applying the 5 steps mentioned in the previous section. Suppose a researcher claims that the mean average weight of men is greater than 100kgs with a standard deviation of 15kgs. 30 men are chosen with an average weight of 112.5 Kgs. Using hypothesis testing, check if there is enough evidence to support the researcher's claim. The confidence interval is given as 95%.

Step 1: This is an example of a right-tailed test. Set up the null hypothesis as \(H_{0}\): \(\mu\) = 100.

Step 2: The alternative hypothesis is given by \(H_{1}\): \(\mu\) > 100.

Step 3: As this is a one-tailed test, \(\alpha\) = 100% - 95% = 5%. This can be used to determine the critical value.

1 - \(\alpha\) = 1 - 0.05 = 0.95

0.95 gives the required area under the curve. Now using a normal distribution table, the area 0.95 is at z = 1.645. A similar process can be followed for a t-test. The only additional requirement is to calculate the degrees of freedom given by n - 1.

Step 4: Calculate the z test statistic. This is because the sample size is 30. Furthermore, the sample and population means are known along with the standard deviation.

z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\).

\(\mu\) = 100, \(\overline{x}\) = 112.5, n = 30, \(\sigma\) = 15

z = \(\frac{112.5-100}{\frac{15}{\sqrt{30}}}\) = 4.56

Step 5: Conclusion. As 4.56 > 1.645 thus, the null hypothesis can be rejected.

Hypothesis Testing and Confidence Intervals

Confidence intervals form an important part of hypothesis testing. This is because the alpha level can be determined from a given confidence interval. Suppose a confidence interval is given as 95%. Subtract the confidence interval from 100%. This gives 100 - 95 = 5% or 0.05. This is the alpha value of a one-tailed hypothesis testing. To obtain the alpha value for a two-tailed hypothesis testing, divide this value by 2. This gives 0.05 / 2 = 0.025.

Related Articles:

• Probability and Statistics
• Data Handling

Important Notes on Hypothesis Testing

• Hypothesis testing is a technique that is used to verify whether the results of an experiment are statistically significant.
• It involves the setting up of a null hypothesis and an alternate hypothesis.
• There are three types of tests that can be conducted under hypothesis testing - z test, t test, and chi square test.
• Hypothesis testing can be classified as right tail, left tail, and two tail tests.

Examples on Hypothesis Testing

• Example 1: The average weight of a dumbbell in a gym is 90lbs. However, a physical trainer believes that the average weight might be higher. A random sample of 5 dumbbells with an average weight of 110lbs and a standard deviation of 18lbs. Using hypothesis testing check if the physical trainer's claim can be supported for a 95% confidence level. Solution: As the sample size is lesser than 30, the t-test is used. \(H_{0}\): \(\mu\) = 90, \(H_{1}\): \(\mu\) > 90 \(\overline{x}\) = 110, \(\mu\) = 90, n = 5, s = 18. \(\alpha\) = 0.05 Using the t-distribution table, the critical value is 2.132 t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\) t = 2.484 As 2.484 > 2.132, the null hypothesis is rejected. Answer: The average weight of the dumbbells may be greater than 90lbs
• Example 2: The average score on a test is 80 with a standard deviation of 10. With a new teaching curriculum introduced it is believed that this score will change. On random testing, the score of 38 students, the mean was found to be 88. With a 0.05 significance level, is there any evidence to support this claim? Solution: This is an example of two-tail hypothesis testing. The z test will be used. \(H_{0}\): \(\mu\) = 80, \(H_{1}\): \(\mu\) ≠ 80 \(\overline{x}\) = 88, \(\mu\) = 80, n = 36, \(\sigma\) = 10. \(\alpha\) = 0.05 / 2 = 0.025 The critical value using the normal distribution table is 1.96 z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\) z = \(\frac{88-80}{\frac{10}{\sqrt{36}}}\) = 4.8 As 4.8 > 1.96, the null hypothesis is rejected. Answer: There is a difference in the scores after the new curriculum was introduced.
• Example 3: The average score of a class is 90. However, a teacher believes that the average score might be lower. The scores of 6 students were randomly measured. The mean was 82 with a standard deviation of 18. With a 0.05 significance level use hypothesis testing to check if this claim is true. Solution: The t test will be used. \(H_{0}\): \(\mu\) = 90, \(H_{1}\): \(\mu\) < 90 \(\overline{x}\) = 110, \(\mu\) = 90, n = 6, s = 18 The critical value from the t table is -2.015 t = \(\frac{\overline{x}-\mu}{\frac{s}{\sqrt{n}}}\) t = \(\frac{82-90}{\frac{18}{\sqrt{6}}}\) t = -1.088 As -1.088 > -2.015, we fail to reject the null hypothesis. Answer: There is not enough evidence to support the claim.

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FAQs on Hypothesis Testing

What is hypothesis testing.

Hypothesis testing in statistics is a tool that is used to make inferences about the population data. It is also used to check if the results of an experiment are valid.

What is the z Test in Hypothesis Testing?

The z test in hypothesis testing is used to find the z test statistic for normally distributed data . The z test is used when the standard deviation of the population is known and the sample size is greater than or equal to 30.

What is the t Test in Hypothesis Testing?

The t test in hypothesis testing is used when the data follows a student t distribution . It is used when the sample size is less than 30 and standard deviation of the population is not known.

What is the formula for z test in Hypothesis Testing?

The formula for a one sample z test in hypothesis testing is z = \(\frac{\overline{x}-\mu}{\frac{\sigma}{\sqrt{n}}}\) and for two samples is z = \(\frac{(\overline{x_{1}}-\overline{x_{2}})-(\mu_{1}-\mu_{2})}{\sqrt{\frac{\sigma_{1}^{2}}{n_{1}}+\frac{\sigma_{2}^{2}}{n_{2}}}}\).

What is the p Value in Hypothesis Testing?

The p value helps to determine if the test results are statistically significant or not. In hypothesis testing, the null hypothesis can either be rejected or not rejected based on the comparison between the p value and the alpha level.

What is One Tail Hypothesis Testing?

When the rejection region is only on one side of the distribution curve then it is known as one tail hypothesis testing. The right tail test and the left tail test are two types of directional hypothesis testing.

What is the Alpha Level in Two Tail Hypothesis Testing?

To get the alpha level in a two tail hypothesis testing divide \(\alpha\) by 2. This is done as there are two rejection regions in the curve.

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Unit 12: Significance tests (hypothesis testing)

About this unit.

Significance tests give us a formal process for using sample data to evaluate the likelihood of some claim about a population value. Learn how to conduct significance tests and calculate p-values to see how likely a sample result is to occur by random chance. You'll also see how we use p-values to make conclusions about hypotheses.

The idea of significance tests

• Simple hypothesis testing (Opens a modal)
• Idea behind hypothesis testing (Opens a modal)
• Examples of null and alternative hypotheses (Opens a modal)
• P-values and significance tests (Opens a modal)
• Comparing P-values to different significance levels (Opens a modal)
• Estimating a P-value from a simulation (Opens a modal)
• Using P-values to make conclusions (Opens a modal)
• Simple hypothesis testing Get 3 of 4 questions to level up!
• Writing null and alternative hypotheses Get 3 of 4 questions to level up!
• Estimating P-values from simulations Get 3 of 4 questions to level up!

Error probabilities and power

• Introduction to Type I and Type II errors (Opens a modal)
• Type 1 errors (Opens a modal)
• Examples identifying Type I and Type II errors (Opens a modal)
• Introduction to power in significance tests (Opens a modal)
• Examples thinking about power in significance tests (Opens a modal)
• Consequences of errors and significance (Opens a modal)
• Type I vs Type II error Get 3 of 4 questions to level up!
• Error probabilities and power Get 3 of 4 questions to level up!

Tests about a population proportion

• Constructing hypotheses for a significance test about a proportion (Opens a modal)
• Conditions for a z test about a proportion (Opens a modal)
• Reference: Conditions for inference on a proportion (Opens a modal)
• Calculating a z statistic in a test about a proportion (Opens a modal)
• Calculating a P-value given a z statistic (Opens a modal)
• Making conclusions in a test about a proportion (Opens a modal)
• Writing hypotheses for a test about a proportion Get 3 of 4 questions to level up!
• Conditions for a z test about a proportion Get 3 of 4 questions to level up!
• Calculating the test statistic in a z test for a proportion Get 3 of 4 questions to level up!
• Calculating the P-value in a z test for a proportion Get 3 of 4 questions to level up!
• Making conclusions in a z test for a proportion Get 3 of 4 questions to level up!

Tests about a population mean

• Writing hypotheses for a significance test about a mean (Opens a modal)
• Conditions for a t test about a mean (Opens a modal)
• Reference: Conditions for inference on a mean (Opens a modal)
• When to use z or t statistics in significance tests (Opens a modal)
• Example calculating t statistic for a test about a mean (Opens a modal)
• Using TI calculator for P-value from t statistic (Opens a modal)
• Using a table to estimate P-value from t statistic (Opens a modal)
• Comparing P-value from t statistic to significance level (Opens a modal)
• Free response example: Significance test for a mean (Opens a modal)
• Writing hypotheses for a test about a mean Get 3 of 4 questions to level up!
• Conditions for a t test about a mean Get 3 of 4 questions to level up!
• Calculating the test statistic in a t test for a mean Get 3 of 4 questions to level up!
• Calculating the P-value in a t test for a mean Get 3 of 4 questions to level up!
• Making conclusions in a t test for a mean Get 3 of 4 questions to level up!

More significance testing videos

• Hypothesis testing and p-values (Opens a modal)
• One-tailed and two-tailed tests (Opens a modal)
• Z-statistics vs. T-statistics (Opens a modal)
• Small sample hypothesis test (Opens a modal)
• Large sample proportion hypothesis testing (Opens a modal)

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• Knowledge Base
• Inferential Statistics | An Easy Introduction & Examples

Inferential Statistics | An Easy Introduction & Examples

Published on September 4, 2020 by Pritha Bhandari . Revised on June 22, 2023.

While descriptive statistics summarize the characteristics of a data set, inferential statistics help you come to conclusions and make predictions based on your data.

When you have collected data from a sample , you can use inferential statistics to understand the larger population from which the sample is taken.

Inferential statistics have two main uses:

• making estimates about populations (for example, the mean SAT score of all 11th graders in the US).
• testing hypotheses to draw conclusions about populations (for example, the relationship between SAT scores and family income).

Table of contents

Descriptive versus inferential statistics, estimating population parameters from sample statistics, hypothesis testing, other interesting articles, frequently asked questions about inferential statistics.

Descriptive statistics allow you to describe a data set, while inferential statistics allow you to make inferences based on a data set.

• Descriptive statistics

Using descriptive statistics, you can report characteristics of your data:

• The distribution concerns the frequency of each value.
• The central tendency concerns the averages of the values.
• The variability concerns how spread out the values are.

In descriptive statistics, there is no uncertainty – the statistics precisely describe the data that you collected. If you collect data from an entire population, you can directly compare these descriptive statistics to those from other populations.

Inferential statistics

Most of the time, you can only acquire data from samples, because it is too difficult or expensive to collect data from the whole population that you’re interested in.

While descriptive statistics can only summarize a sample’s characteristics, inferential statistics use your sample to make reasonable guesses about the larger population.

With inferential statistics, it’s important to use random and unbiased sampling methods . If your sample isn’t representative of your population, then you can’t make valid statistical inferences or generalize .

Sampling error in inferential statistics

Since the size of a sample is always smaller than the size of the population, some of the population isn’t captured by sample data. This creates sampling error , which is the difference between the true population values (called parameters) and the measured sample values (called statistics).

Sampling error arises any time you use a sample, even if your sample is random and unbiased. For this reason, there is always some uncertainty in inferential statistics. However, using probability sampling methods reduces this uncertainty.

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The characteristics of samples and populations are described by numbers called statistics and parameters :

• A statistic is a measure that describes the sample (e.g., sample mean ).
• A parameter is a measure that describes the whole population (e.g., population mean).

Sampling error is the difference between a parameter and a corresponding statistic. Since in most cases you don’t know the real population parameter, you can use inferential statistics to estimate these parameters in a way that takes sampling error into account.

There are two important types of estimates you can make about the population: point estimates and interval estimates .

• A point estimate is a single value estimate of a parameter. For instance, a sample mean is a point estimate of a population mean.
• An interval estimate gives you a range of values where the parameter is expected to lie. A confidence interval is the most common type of interval estimate.

Both types of estimates are important for gathering a clear idea of where a parameter is likely to lie.

Confidence intervals

A confidence interval uses the variability around a statistic to come up with an interval estimate for a parameter. Confidence intervals are useful for estimating parameters because they take sampling error into account.

While a point estimate gives you a precise value for the parameter you are interested in, a confidence interval tells you the uncertainty of the point estimate. They are best used in combination with each other.

Each confidence interval is associated with a confidence level. A confidence level tells you the probability (in percentage) of the interval containing the parameter estimate if you repeat the study again.

A 95% confidence interval means that if you repeat your study with a new sample in exactly the same way 100 times, you can expect your estimate to lie within the specified range of values 95 times.

Although you can say that your estimate will lie within the interval a certain percentage of the time, you cannot say for sure that the actual population parameter will. That’s because you can’t know the true value of the population parameter without collecting data from the full population.

However, with random sampling and a suitable sample size, you can reasonably expect your confidence interval to contain the parameter a certain percentage of the time.

Your point estimate of the population mean paid vacation days is the sample mean of 19 paid vacation days.

Hypothesis testing is a formal process of statistical analysis using inferential statistics. The goal of hypothesis testing is to compare populations or assess relationships between variables using samples.

Hypotheses , or predictions, are tested using statistical tests . Statistical tests also estimate sampling errors so that valid inferences can be made.

Statistical tests can be parametric or non-parametric. Parametric tests are considered more statistically powerful because they are more likely to detect an effect if one exists.

Parametric tests make assumptions that include the following:

• the population that the sample comes from follows a normal distribution of scores
• the sample size is large enough to represent the population
• the variances , a measure of variability , of each group being compared are similar

When your data violates any of these assumptions, non-parametric tests are more suitable. Non-parametric tests are called “distribution-free tests” because they don’t assume anything about the distribution of the population data.

Statistical tests come in three forms: tests of comparison, correlation or regression.

Comparison tests

Comparison tests assess whether there are differences in means, medians or rankings of scores of two or more groups.

To decide which test suits your aim, consider whether your data meets the conditions necessary for parametric tests, the number of samples, and the levels of measurement of your variables.

Means can only be found for interval or ratio data , while medians and rankings are more appropriate measures for ordinal data .

Correlation tests

Correlation tests determine the extent to which two variables are associated.

Although Pearson’s r is the most statistically powerful test, Spearman’s r is appropriate for interval and ratio variables when the data doesn’t follow a normal distribution.

The chi square test of independence is the only test that can be used with nominal variables.

Regression tests

Regression tests demonstrate whether changes in predictor variables cause changes in an outcome variable. You can decide which regression test to use based on the number and types of variables you have as predictors and outcomes.

Most of the commonly used regression tests are parametric. If your data is not normally distributed, you can perform data transformations.

Data transformations help you make your data normally distributed using mathematical operations, like taking the square root of each value.

If you want to know more about statistics , methodology , or research bias , make sure to check out some of our other articles with explanations and examples.

• Confidence interval
• Measures of central tendency
• Correlation coefficient

Methodology

• Cluster sampling
• Stratified sampling
• Types of interviews
• Cohort study
• Thematic analysis

Research bias

• Implicit bias
• Cognitive bias
• Survivorship bias
• Availability heuristic
• Nonresponse bias
• Regression to the mean

Descriptive statistics summarize the characteristics of a data set. Inferential statistics allow you to test a hypothesis or assess whether your data is generalizable to the broader population.

A statistic refers to measures about the sample , while a parameter refers to measures about the population .

A sampling error is the difference between a population parameter and a sample statistic .

Hypothesis testing is a formal procedure for investigating our ideas about the world using statistics. It is used by scientists to test specific predictions, called hypotheses , by calculating how likely it is that a pattern or relationship between variables could have arisen by chance.

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Hypothesis testing

When interpreting research findings, researchers need to assess whether these findings may have occurred by chance. Hypothesis testing is a systematic procedure for deciding whether the results of a research study support a particular theory which applies to a population.

Hypothesis testing uses sample data to evaluate a hypothesis about a population . A hypothesis test assesses how unusual the result is, whether it is reasonable chance variation or whether the result is too extreme to be considered chance variation.

Basic concepts

• Null and research hypothesis

Probability value and types of errors

Effect size and statistical significance.

• Directional and non-directional hypotheses

Null and research hypotheses

To carry out statistical hypothesis testing, research and null hypothesis are employed:

• Research hypothesis : this is the hypothesis that you propose, also known as the alternative hypothesis HA. For example:

H A: There is a relationship between intelligence and academic results.

H A: First year university students obtain higher grades after an intensive Statistics course.

H A; Males and females differ in their levels of stress.

• The null hypothesis (H o ) is the opposite of the research hypothesis and expresses that there is no relationship between variables, or no differences between groups; for example:

H o : There is no relationship between intelligence and academic results.

H o:  First year university students do not obtain higher grades after an intensive Statistics course.

H o : Males and females will not differ in their levels of stress.

The purpose of hypothesis testing is to test whether the null hypothesis (there is no difference, no effect) can be rejected or approved. If the null hypothesis is rejected, then the research hypothesis can be accepted. If the null hypothesis is accepted, then the research hypothesis is rejected.

In hypothesis testing, a value is set to assess whether the null hypothesis is accepted or rejected and whether the result is statistically significant:

• A critical value is the score the sample would need to decide against the null hypothesis.
• A probability value is used to assess the significance of the statistical test. If the null hypothesis is rejected, then the alternative to the null hypothesis is accepted.

The probability value, or p value , is the probability of an outcome or research result given the hypothesis. Usually, the probability value is set at 0.05: the null hypothesis will be rejected if the probability value of the statistical test is less than 0.05. There are two types of errors associated to hypothesis testing:

• What if we observe a difference – but none exists in the population?
• What if we do not find a difference – but it does exist in the population?

These situations are known as Type I and Type II errors:

• Type I Error: is the type of error that involves the rejection of a null hypothesis that is actually true (i.e. a false positive).
• Type II Error:  is the type of error that occurs when we do not reject a null hypothesis that is false (i.e. a false negative).

These errors cannot be eliminated; they can be minimised, but minimising one type of error will increase the probability of committing the other type.

The probability of making a Type I error depends on the criterion that is used to accept or reject the null hypothesis: the p value or alpha level . The alpha is set by the researcher, usually at .05, and is the chance the researcher is willing to take and still claim the significance of the statistical test.). Choosing a smaller alpha level will decrease the likelihood of committing Type I error.

For example, p<0.05  indicates that there are 5 chances in 100 that the difference observed was really due to sampling error – that 5% of the time a Type I error will occur or that there is a 5% chance that the opposite of the null hypothesis is actually true.

With a p<0.01, there will be 1 chance in 100 that the difference observed was really due to sampling error – 1% of the time a Type I error will occur.

The p level is specified before analysing the data. If the data analysis results in a probability value below the α (alpha) level, then the null hypothesis is rejected; if it is not, then the null hypothesis is not rejected.

When the null hypothesis is rejected, the effect is said to be statistically significant. However, statistical significance does not mean that the effect is important.

A result can be statistically significant, but the effect size may be small. Finding that an effect is significant does not provide information about how large or important the effect is. In fact, a small effect can be statistically significant if the sample size is large enough.

Information about the effect size, or magnitude of the result, is given by the statistical test. For example, the strength of the correlation between two variables is given by the coefficient of correlation, which varies from 0 to 1.

• A hypothesis that states that students who attend an intensive Statistics course will obtain higher grades than students who do not attend would be directional.
• A non-directional hypothesis states that there will be differences between students who attend do or don’t attend an intensive Statistics course, but we don’t know what group will get higher grades than the other. The hypothesis only states that they will obtain different grades.

The hypothesis testing process

The hypothesis testing process can be divided into five steps:

• Restate the research question as research hypothesis and a null hypothesis about the populations.
• Determine the characteristics of the comparison distribution.
• Determine the cut off sample score on the comparison distribution at which the null hypothesis should be rejected.
• Determine your sample’s score on the comparison distribution.
• Decide whether to reject the null hypothesis.

This example illustrates how these five steps can be applied to text a hypothesis:

• Let’s say that you conduct an experiment to investigate whether students’ ability to memorise words improves after they have consumed caffeine.
• The experiment involves two groups of students: the first group consumes caffeine; the second group drinks water.
• Both groups complete a memory test.
• A randomly selected individual in the experimental condition (i.e. the group that consumes caffeine) has a score of 27 on the memory test. The scores of people in general on this memory measure are normally distributed with a mean of 19 and a standard deviation of 4.
• The researcher predicts an effect (differences in memory for these groups) but does not predict a particular direction of effect (i.e. which group will have higher scores on the memory test). Using the 5% significance level, what should you conclude?

Step 1 : There are two populations of interest.

Population 1: People who go through the experimental procedure (drink coffee).

Population 2: People who do not go through the experimental procedure (drink water).

• Research hypothesis: Population 1 will score differently from Population 2.
• Null hypothesis: There will be no difference between the two populations.

Step 2 : We know that the characteristics of the comparison distribution (student population) are:

Population M = 19, Population SD= 4, normally distributed. These are the mean and standard deviation of the distribution of scores on the memory test for the general student population.

Step 3 : For a two-tailed test (the direction of the effect is not specified) at the 5% level (25% at each tail), the cut off sample scores are +1.96 and -1.99.

Step 4 : Your sample score of 27 needs to be converted into a Z value. To calculate Z = (27-19)/4= 2 ( check the Converting into Z scores section if you need to review how to do this process)

Step 5 : A ‘Z’ score of 2 is more extreme than the cut off Z of +1.96 (see figure above). The result is significant and, thus, the null hypothesis is rejected.

You can find more examples here:

• Statistics (RMIT Learning Lab)

Some commonly used statistical techniques

Correlation analysis, multiple regression.

• Analysis of variance

Chi-square test for independence

Correlation analysis explores the association between variables . The purpose of correlational analysis is to discover whether there is a relationship between variables, which is unlikely to occur by sampling error. The null hypothesis is that there is no relationship between the two variables. Correlation analysis provides information about:

• The direction of the relationship: positive or negative- given by the sign of the correlation coefficient.
• The strength or magnitude of the relationship between the two variables- given by the correlation coefficient, which varies from 0 (no relationship between the variables) to 1 (perfect relationship between the variables).
• Direction of the relationship.

A positive correlation indicates that high scores on one variable are associated with high scores on the other variable; low scores on one variable are associated with low scores on the second variable . For instance, in the figure below, higher scores on negative affect are associated with higher scores on perceived stress

A negative correlation indicates that high scores on one variable are associated with low scores on the other variable. The graph shows that a person who scores high on perceived stress will probably score low on mastery. The slope of the graph is downwards- as it moves to the right. In the figure below, higher scores on mastery are associated with lower scores on perceived stress.

Fig 2. Negative correlation between two variables. Adapted from Pallant, J. (2013). SPSS survival manual: A step by step guide to data analysis using IBM SPSS (5th ed.). Sydney, Melbourne, Auckland, London: Allen & Unwin

2. The strength or magnitude of the relationship

The strength of a linear relationship between two variables is measured by a statistic known as the correlation coefficient , which varies from 0 to -1, and from 0 to +1. There are several correlation coefficients; the most widely used are Pearson’s r and Spearman’s rho. The strength of the relationship is interpreted as follows:

• Small/weak: r= .10 to .29
• Medium/moderate: r= .30 to .49
• Large/strong: r= .50 to 1

It is important to note that correlation analysis does not imply causality. Correlation is used to explore the association between variables, however, it does not indicate that one variable causes the other. The correlation between two variables could be due to the fact that a third variable is affecting the two variables.

Multiple regression is an extension of correlation analysis. Multiple regression is used to explore the relationship between one dependent variable and a number of independent variables or predictors . The purpose of a multiple regression model is to predict values of a dependent variable based on the values of the independent variables or predictors. For example, a researcher may be interested in predicting students’ academic success (e.g. grades) based on a number of predictors, for example, hours spent studying, satisfaction with studies, relationships with peers and lecturers.

A multiple regression model can be conducted using statistical software (e.g. SPSS). The software will test the significance of the model (i.e. does the model significantly predicts scores on the dependent variable using the independent variables introduced in the model?), how much of the variance in the dependent variable is explained by the model, and the individual contribution of each independent variable.

Example of multiple regression model

From Dunn et al. (2014). Influence of academic self-regulation, critical thinking, and age on online graduate students' academic help-seeking.

In this model, help-seeking is the dependent variable; there are three independent variables or predictors. The coefficients show the direction (positive or negative) and magnitude of the relationship between each predictor and the dependent variable. The model was statistically significant and predicted 13.5% of the variance in help-seeking.

t-Tests are employed to compare the mean score on some continuous variable for two groups . The null hypothesis to be tested is there are no differences between the two groups (e.g. anxiety scores for males and females are not different).

If the significance value of the t-test is equal or less than .05, there is a significant difference in the mean scores on the variable of interest for each of the two groups. If the value is above .05, there is no significant difference between the groups.

t-Tests can be employed to compare the mean scores of two different groups (independent-samples t-test ) or to compare the same group of people on two different occasions ( paired-samples t-test) .

In addition to assessing whether the difference between the two groups is statistically significant, it is important to consider the effect size or magnitude of the difference between the groups. The effect size is given by partial eta squared (proportion of variance of the dependent variable that is explained by the independent variable) and Cohen’s d (difference between groups in terms of standard deviation units).

In this example, an independent samples t-test was conducted to assess whether males and females differ in their perceived anxiety levels. The significance of the test is .004. Since this value is less than .05, we can conclude that there is a statistically significant difference between males and females in their perceived anxiety levels.

Whilst t-tests compare the mean score on one variable for two groups, analysis of variance is used to test more than two groups . Following the previous example, analysis of variance would be employed to test whether there are differences in anxiety scores for students from different disciplines.

Analysis of variance compare the variance (variability in scores) between the different groups (believed to be due to the independent variable) with the variability within each group (believed to be due to chance). An F ratio is calculated; a large F ratio indicates that there is more variability between the groups (caused by the independent variable) than there is within each group (error term). A significant F test indicates that we can reject the null hypothesis; i.e. that there is no difference between the groups.

Again, effect size statistics such as Cohen’s d and eta squared are employed to assess the magnitude of the differences between groups.

In this example, we examined differences in perceived anxiety between students from different disciplines. The results of the Anova Test show that the significance level is .005. Since this value is below .05, we can conclude that there are statistically significant differences between students from different disciplines in their perceived anxiety levels.

Chi-square test for independence is used to explore the relationship between two categorical variables. Each variable can have two or more categories.

For example, a researcher can use a Chi-square test for independence to assess the relationship between study disciplines (e.g. Psychology, Business, Education,…) and help-seeking behaviour (Yes/No). The test compares the observed frequencies of cases with the values that would be expected if there was no association between the two variables of interest. A statistically significant Chi-square test indicates that the two variables are associated (e.g. Psychology students are more likely to seek help than Business students). The effect size is assessed using effect size statistics: Phi and Cramer’s V .

In this example, a Chi-square test was conducted to assess whether males and females differ in their help-seeking behaviour (Yes/No). The crosstabulation table shows the percentage of males of females who sought/didn't seek help. The table 'Chi square tests' shows the significance of the test (Pearson Chi square asymp sig: .482). Since this value is above .05, we conclude that there is no statistically significant difference between males and females in their help-seeking behaviour.

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Lesson 10 of 24 By Avijeet Biswal

Table of Contents

In today’s data-driven world, decisions are based on data all the time. Hypothesis plays a crucial role in that process, whether it may be making business decisions, in the health sector, academia, or in quality improvement. Without hypothesis & hypothesis tests, you risk drawing the wrong conclusions and making bad decisions. In this tutorial, you will look at Hypothesis Testing in Statistics.

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What Is Hypothesis Testing in Statistics?

Hypothesis Testing is a type of statistical analysis in which you put your assumptions about a population parameter to the test. It is used to estimate the relationship between 2 statistical variables.

Let's discuss few examples of statistical hypothesis from real-life - 

• A teacher assumes that 60% of his college's students come from lower-middle-class families.
• A doctor believes that 3D (Diet, Dose, and Discipline) is 90% effective for diabetic patients.

Now that you know about hypothesis testing, look at the two types of hypothesis testing in statistics.

Hypothesis Testing Formula

Z = ( x̅ – μ0 ) / (σ /√n)

• Here, x̅ is the sample mean,
• μ0 is the population mean,
• σ is the standard deviation,
• n is the sample size.

How Hypothesis Testing Works?

An analyst performs hypothesis testing on a statistical sample to present evidence of the plausibility of the null hypothesis. Measurements and analyses are conducted on a random sample of the population to test a theory. Analysts use a random population sample to test two hypotheses: the null and alternative hypotheses.

The null hypothesis is typically an equality hypothesis between population parameters; for example, a null hypothesis may claim that the population means return equals zero. The alternate hypothesis is essentially the inverse of the null hypothesis (e.g., the population means the return is not equal to zero). As a result, they are mutually exclusive, and only one can be correct. One of the two possibilities, however, will always be correct.

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Null Hypothesis and Alternate Hypothesis

The Null Hypothesis is the assumption that the event will not occur. A null hypothesis has no bearing on the study's outcome unless it is rejected.

H0 is the symbol for it, and it is pronounced H-naught.

The Alternate Hypothesis is the logical opposite of the null hypothesis. The acceptance of the alternative hypothesis follows the rejection of the null hypothesis. H1 is the symbol for it.

Let's understand this with an example.

A sanitizer manufacturer claims that its product kills 95 percent of germs on average. 

To put this company's claim to the test, create a null and alternate hypothesis.

H0 (Null Hypothesis): Average = 95%.

Alternative Hypothesis (H1): The average is less than 95%.

Another straightforward example to understand this concept is determining whether or not a coin is fair and balanced. The null hypothesis states that the probability of a show of heads is equal to the likelihood of a show of tails. In contrast, the alternate theory states that the probability of a show of heads and tails would be very different.

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Hypothesis Testing Calculation With Examples

Let's consider a hypothesis test for the average height of women in the United States. Suppose our null hypothesis is that the average height is 5'4". We gather a sample of 100 women and determine that their average height is 5'5". The standard deviation of population is 2.

To calculate the z-score, we would use the following formula:

z = ( x̅ – μ0 ) / (σ /√n)

z = (5'5" - 5'4") / (2" / √100)

z = 0.5 / (0.045)

We will reject the null hypothesis as the z-score of 11.11 is very large and conclude that there is evidence to suggest that the average height of women in the US is greater than 5'4".

Steps of Hypothesis Testing

Hypothesis testing is a statistical method to determine if there is enough evidence in a sample of data to infer that a certain condition is true for the entire population. Here’s a breakdown of the typical steps involved in hypothesis testing:

Formulate Hypotheses

• Null Hypothesis (H0): This hypothesis states that there is no effect or difference, and it is the hypothesis you attempt to reject with your test.
• Alternative Hypothesis (H1 or Ha): This hypothesis is what you might believe to be true or hope to prove true. It is usually considered the opposite of the null hypothesis.

Choose the Significance Level (α)

The significance level, often denoted by alpha (α), is the probability of rejecting the null hypothesis when it is true. Common choices for α are 0.05 (5%), 0.01 (1%), and 0.10 (10%).

Select the Appropriate Test

Choose a statistical test based on the type of data and the hypothesis. Common tests include t-tests, chi-square tests, ANOVA, and regression analysis. The selection depends on data type, distribution, sample size, and whether the hypothesis is one-tailed or two-tailed.

Collect Data

Gather the data that will be analyzed in the test. This data should be representative of the population to infer conclusions accurately.

Calculate the Test Statistic

Based on the collected data and the chosen test, calculate a test statistic that reflects how much the observed data deviates from the null hypothesis.

Determine the p-value

The p-value is the probability of observing test results at least as extreme as the results observed, assuming the null hypothesis is correct. It helps determine the strength of the evidence against the null hypothesis.

Make a Decision

Compare the p-value to the chosen significance level:

• If the p-value ≤ α: Reject the null hypothesis, suggesting sufficient evidence in the data supports the alternative hypothesis.
• If the p-value > α: Do not reject the null hypothesis, suggesting insufficient evidence to support the alternative hypothesis.

Report the Results

Present the findings from the hypothesis test, including the test statistic, p-value, and the conclusion about the hypotheses.

Perform Post-hoc Analysis (if necessary)

Depending on the results and the study design, further analysis may be needed to explore the data more deeply or to address multiple comparisons if several hypotheses were tested simultaneously.

Types of Hypothesis Testing

To determine whether a discovery or relationship is statistically significant, hypothesis testing uses a z-test. It usually checks to see if two means are the same (the null hypothesis). Only when the population standard deviation is known and the sample size is 30 data points or more, can a z-test be applied.

A statistical test called a t-test is employed to compare the means of two groups. To determine whether two groups differ or if a procedure or treatment affects the population of interest, it is frequently used in hypothesis testing.

Chi-Square 

You utilize a Chi-square test for hypothesis testing concerning whether your data is as predicted. To determine if the expected and observed results are well-fitted, the Chi-square test analyzes the differences between categorical variables from a random sample. The test's fundamental premise is that the observed values in your data should be compared to the predicted values that would be present if the null hypothesis were true.

Hypothesis Testing and Confidence Intervals

Both confidence intervals and hypothesis tests are inferential techniques that depend on approximating the sample distribution. Data from a sample is used to estimate a population parameter using confidence intervals. Data from a sample is used in hypothesis testing to examine a given hypothesis. We must have a postulated parameter to conduct hypothesis testing.

Bootstrap distributions and randomization distributions are created using comparable simulation techniques. The observed sample statistic is the focal point of a bootstrap distribution, whereas the null hypothesis value is the focal point of a randomization distribution.

A variety of feasible population parameter estimates are included in confidence ranges. In this lesson, we created just two-tailed confidence intervals. There is a direct connection between these two-tail confidence intervals and these two-tail hypothesis tests. The results of a two-tailed hypothesis test and two-tailed confidence intervals typically provide the same results. In other words, a hypothesis test at the 0.05 level will virtually always fail to reject the null hypothesis if the 95% confidence interval contains the predicted value. A hypothesis test at the 0.05 level will nearly certainly reject the null hypothesis if the 95% confidence interval does not include the hypothesized parameter.

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Simple and Composite Hypothesis Testing

Depending on the population distribution, you can classify the statistical hypothesis into two types.

Simple Hypothesis: A simple hypothesis specifies an exact value for the parameter.

Composite Hypothesis: A composite hypothesis specifies a range of values.

A company is claiming that their average sales for this quarter are 1000 units. This is an example of a simple hypothesis.

Suppose the company claims that the sales are in the range of 900 to 1000 units. Then this is a case of a composite hypothesis.

One-Tailed and Two-Tailed Hypothesis Testing

The One-Tailed test, also called a directional test, considers a critical region of data that would result in the null hypothesis being rejected if the test sample falls into it, inevitably meaning the acceptance of the alternate hypothesis.

In a one-tailed test, the critical distribution area is one-sided, meaning the test sample is either greater or lesser than a specific value.

In two tails, the test sample is checked to be greater or less than a range of values in a Two-Tailed test, implying that the critical distribution area is two-sided.

If the sample falls within this range, the alternate hypothesis will be accepted, and the null hypothesis will be rejected.

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Right Tailed Hypothesis Testing

If the larger than (>) sign appears in your hypothesis statement, you are using a right-tailed test, also known as an upper test. Or, to put it another way, the disparity is to the right. For instance, you can contrast the battery life before and after a change in production. Your hypothesis statements can be the following if you want to know if the battery life is longer than the original (let's say 90 hours):

• The null hypothesis is (H0 <= 90) or less change.
• A possibility is that battery life has risen (H1) > 90.

The crucial point in this situation is that the alternate hypothesis (H1), not the null hypothesis, decides whether you get a right-tailed test.

Left Tailed Hypothesis Testing

Alternative hypotheses that assert the true value of a parameter is lower than the null hypothesis are tested with a left-tailed test; they are indicated by the asterisk "<".

Suppose H0: mean = 50 and H1: mean not equal to 50

According to the H1, the mean can be greater than or less than 50. This is an example of a Two-tailed test.

In a similar manner, if H0: mean >=50, then H1: mean <50

Here the mean is less than 50. It is called a One-tailed test.

Type 1 and Type 2 Error

A hypothesis test can result in two types of errors.

Type 1 Error: A Type-I error occurs when sample results reject the null hypothesis despite being true.

Type 2 Error: A Type-II error occurs when the null hypothesis is not rejected when it is false, unlike a Type-I error.

Suppose a teacher evaluates the examination paper to decide whether a student passes or fails.

H0: Student has passed

H1: Student has failed

Type I error will be the teacher failing the student [rejects H0] although the student scored the passing marks [H0 was true]. 

Type II error will be the case where the teacher passes the student [do not reject H0] although the student did not score the passing marks [H1 is true].

Level of Significance

The alpha value is a criterion for determining whether a test statistic is statistically significant. In a statistical test, Alpha represents an acceptable probability of a Type I error. Because alpha is a probability, it can be anywhere between 0 and 1. In practice, the most commonly used alpha values are 0.01, 0.05, and 0.1, which represent a 1%, 5%, and 10% chance of a Type I error, respectively (i.e. rejecting the null hypothesis when it is in fact correct).

A p-value is a metric that expresses the likelihood that an observed difference could have occurred by chance. As the p-value decreases the statistical significance of the observed difference increases. If the p-value is too low, you reject the null hypothesis.

Here you have taken an example in which you are trying to test whether the new advertising campaign has increased the product's sales. The p-value is the likelihood that the null hypothesis, which states that there is no change in the sales due to the new advertising campaign, is true. If the p-value is .30, then there is a 30% chance that there is no increase or decrease in the product's sales.  If the p-value is 0.03, then there is a 3% probability that there is no increase or decrease in the sales value due to the new advertising campaign. As you can see, the lower the p-value, the chances of the alternate hypothesis being true increases, which means that the new advertising campaign causes an increase or decrease in sales.

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Why Is Hypothesis Testing Important in Research Methodology?

Hypothesis testing is crucial in research methodology for several reasons:

• Provides evidence-based conclusions: It allows researchers to make objective conclusions based on empirical data, providing evidence to support or refute their research hypotheses.
• Supports decision-making: It helps make informed decisions, such as accepting or rejecting a new treatment, implementing policy changes, or adopting new practices.
• Adds rigor and validity: It adds scientific rigor to research using statistical methods to analyze data, ensuring that conclusions are based on sound statistical evidence.
• Contributes to the advancement of knowledge: By testing hypotheses, researchers contribute to the growth of knowledge in their respective fields by confirming existing theories or discovering new patterns and relationships.

When Did Hypothesis Testing Begin?

Hypothesis testing as a formalized process began in the early 20th century, primarily through the work of statisticians such as Ronald A. Fisher, Jerzy Neyman, and Egon Pearson. The development of hypothesis testing is closely tied to the evolution of statistical methods during this period.

• Ronald A. Fisher (1920s): Fisher was one of the key figures in developing the foundation for modern statistical science. In the 1920s, he introduced the concept of the null hypothesis in his book "Statistical Methods for Research Workers" (1925). Fisher also developed significance testing to examine the likelihood of observing the collected data if the null hypothesis were true. He introduced p-values to determine the significance of the observed results.
• Neyman-Pearson Framework (1930s): Jerzy Neyman and Egon Pearson built on Fisher’s work and formalized the process of hypothesis testing even further. In the 1930s, they introduced the concepts of Type I and Type II errors and developed a decision-making framework widely used in hypothesis testing today. Their approach emphasized the balance between these errors and introduced the concepts of the power of a test and the alternative hypothesis.

The dialogue between Fisher's and Neyman-Pearson's approaches shaped the methods and philosophy of statistical hypothesis testing used today. Fisher emphasized the evidential interpretation of the p-value. At the same time, Neyman and Pearson advocated for a decision-theoretical approach in which hypotheses are either accepted or rejected based on pre-determined significance levels and power considerations.

The application and methodology of hypothesis testing have since become a cornerstone of statistical analysis across various scientific disciplines, marking a significant statistical development.

Limitations of Hypothesis Testing

Hypothesis testing has some limitations that researchers should be aware of:

• It cannot prove or establish the truth: Hypothesis testing provides evidence to support or reject a hypothesis, but it cannot confirm the absolute truth of the research question.
• Results are sample-specific: Hypothesis testing is based on analyzing a sample from a population, and the conclusions drawn are specific to that particular sample.
• Possible errors: During hypothesis testing, there is a chance of committing type I error (rejecting a true null hypothesis) or type II error (failing to reject a false null hypothesis).
• Assumptions and requirements: Different tests have specific assumptions and requirements that must be met to accurately interpret results.

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After reading this tutorial, you would have a much better understanding of hypothesis testing, one of the most important concepts in the field of Data Science . The majority of hypotheses are based on speculation about observed behavior, natural phenomena, or established theories.

If you are interested in statistics of data science and skills needed for such a career, you ought to explore the Post Graduate Program in Data Science.

If you have any questions regarding this ‘Hypothesis Testing In Statistics’ tutorial, do share them in the comment section. Our subject matter expert will respond to your queries. Happy learning!

1. What is hypothesis testing in statistics with example?

Hypothesis testing is a statistical method used to determine if there is enough evidence in a sample data to draw conclusions about a population. It involves formulating two competing hypotheses, the null hypothesis (H0) and the alternative hypothesis (Ha), and then collecting data to assess the evidence. An example: testing if a new drug improves patient recovery (Ha) compared to the standard treatment (H0) based on collected patient data.

2. What is H0 and H1 in statistics?

In statistics, H0​ and H1​ represent the null and alternative hypotheses. The null hypothesis, H0​, is the default assumption that no effect or difference exists between groups or conditions. The alternative hypothesis, H1​, is the competing claim suggesting an effect or a difference. Statistical tests determine whether to reject the null hypothesis in favor of the alternative hypothesis based on the data.

3. What is a simple hypothesis with an example?

A simple hypothesis is a specific statement predicting a single relationship between two variables. It posits a direct and uncomplicated outcome. For example, a simple hypothesis might state, "Increased sunlight exposure increases the growth rate of sunflowers." Here, the hypothesis suggests a direct relationship between the amount of sunlight (independent variable) and the growth rate of sunflowers (dependent variable), with no additional variables considered.

4. What are the 2 types of hypothesis testing?

• One-tailed (or one-sided) test: Tests for the significance of an effect in only one direction, either positive or negative.
• Two-tailed (or two-sided) test: Tests for the significance of an effect in both directions, allowing for the possibility of a positive or negative effect.

The choice between one-tailed and two-tailed tests depends on the specific research question and the directionality of the expected effect.

5. What are the 3 major types of hypothesis?

The three major types of hypotheses are:

• Null Hypothesis (H0): Represents the default assumption, stating that there is no significant effect or relationship in the data.
• Alternative Hypothesis (Ha): Contradicts the null hypothesis and proposes a specific effect or relationship that researchers want to investigate.
• Nondirectional Hypothesis: An alternative hypothesis that doesn't specify the direction of the effect, leaving it open for both positive and negative possibilities.

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About the Author

Avijeet is a Senior Research Analyst at Simplilearn. Passionate about Data Analytics, Machine Learning, and Deep Learning, Avijeet is also interested in politics, cricket, and football.

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What Is Hypothesis Testing?

• How It Works

4 Step Process

The bottom line.

• Fundamental Analysis

Hypothesis Testing: 4 Steps and Example

Hypothesis testing, sometimes called significance testing, is an act in statistics whereby an analyst tests an assumption regarding a population parameter. The methodology employed by the analyst depends on the nature of the data used and the reason for the analysis.

Hypothesis testing is used to assess the plausibility of a hypothesis by using sample data. Such data may come from a larger population or a data-generating process. The word "population" will be used for both of these cases in the following descriptions.

Key Takeaways

• Hypothesis testing is used to assess the plausibility of a hypothesis by using sample data.
• The test provides evidence concerning the plausibility of the hypothesis, given the data.
• Statistical analysts test a hypothesis by measuring and examining a random sample of the population being analyzed.
• The four steps of hypothesis testing include stating the hypotheses, formulating an analysis plan, analyzing the sample data, and analyzing the result.

How Hypothesis Testing Works

In hypothesis testing, an  analyst  tests a statistical sample, intending to provide evidence on the plausibility of the null hypothesis. Statistical analysts measure and examine a random sample of the population being analyzed. All analysts use a random population sample to test two different hypotheses: the null hypothesis and the alternative hypothesis.

The null hypothesis is usually a hypothesis of equality between population parameters; e.g., a null hypothesis may state that the population mean return is equal to zero. The alternative hypothesis is effectively the opposite of a null hypothesis. Thus, they are mutually exclusive , and only one can be true. However, one of the two hypotheses will always be true.

The null hypothesis is a statement about a population parameter, such as the population mean, that is assumed to be true.

• State the hypotheses.
• Formulate an analysis plan, which outlines how the data will be evaluated.
• Carry out the plan and analyze the sample data.
• Analyze the results and either reject the null hypothesis, or state that the null hypothesis is plausible, given the data.

Example of Hypothesis Testing

If an individual wants to test that a penny has exactly a 50% chance of landing on heads, the null hypothesis would be that 50% is correct, and the alternative hypothesis would be that 50% is not correct. Mathematically, the null hypothesis is represented as Ho: P = 0.5. The alternative hypothesis is shown as "Ha" and is identical to the null hypothesis, except with the equal sign struck-through, meaning that it does not equal 50%.

A random sample of 100 coin flips is taken, and the null hypothesis is tested. If it is found that the 100 coin flips were distributed as 40 heads and 60 tails, the analyst would assume that a penny does not have a 50% chance of landing on heads and would reject the null hypothesis and accept the alternative hypothesis.

If there were 48 heads and 52 tails, then it is plausible that the coin could be fair and still produce such a result. In cases such as this where the null hypothesis is "accepted," the analyst states that the difference between the expected results (50 heads and 50 tails) and the observed results (48 heads and 52 tails) is "explainable by chance alone."

When Did Hypothesis Testing Begin?

Some statisticians attribute the first hypothesis tests to satirical writer John Arbuthnot in 1710, who studied male and female births in England after observing that in nearly every year, male births exceeded female births by a slight proportion. Arbuthnot calculated that the probability of this happening by chance was small, and therefore it was due to “divine providence.”

What are the Benefits of Hypothesis Testing?

Hypothesis testing helps assess the accuracy of new ideas or theories by testing them against data. This allows researchers to determine whether the evidence supports their hypothesis, helping to avoid false claims and conclusions. Hypothesis testing also provides a framework for decision-making based on data rather than personal opinions or biases. By relying on statistical analysis, hypothesis testing helps to reduce the effects of chance and confounding variables, providing a robust framework for making informed conclusions.

What are the Limitations of Hypothesis Testing?

Hypothesis testing relies exclusively on data and doesn’t provide a comprehensive understanding of the subject being studied. Additionally, the accuracy of the results depends on the quality of the available data and the statistical methods used. Inaccurate data or inappropriate hypothesis formulation may lead to incorrect conclusions or failed tests. Hypothesis testing can also lead to errors, such as analysts either accepting or rejecting a null hypothesis when they shouldn’t have. These errors may result in false conclusions or missed opportunities to identify significant patterns or relationships in the data.

Hypothesis testing refers to a statistical process that helps researchers determine the reliability of a study. By using a well-formulated hypothesis and set of statistical tests, individuals or businesses can make inferences about the population that they are studying and draw conclusions based on the data presented. All hypothesis testing methods have the same four-step process, which includes stating the hypotheses, formulating an analysis plan, analyzing the sample data, and analyzing the result.

Sage. " Introduction to Hypothesis Testing ," Page 4.

Elder Research. " Who Invented the Null Hypothesis? "

Formplus. " Hypothesis Testing: Definition, Uses, Limitations and Examples ."

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Home » What is a Hypothesis – Types, Examples and Writing Guide

What is a Hypothesis – Types, Examples and Writing Guide

Table of Contents

Definition:

Hypothesis is an educated guess or proposed explanation for a phenomenon, based on some initial observations or data. It is a tentative statement that can be tested and potentially proven or disproven through further investigation and experimentation.

Hypothesis is often used in scientific research to guide the design of experiments and the collection and analysis of data. It is an essential element of the scientific method, as it allows researchers to make predictions about the outcome of their experiments and to test those predictions to determine their accuracy.

Types of Hypothesis

Types of Hypothesis are as follows:

Research Hypothesis

A research hypothesis is a statement that predicts a relationship between variables. It is usually formulated as a specific statement that can be tested through research, and it is often used in scientific research to guide the design of experiments.

Null Hypothesis

The null hypothesis is a statement that assumes there is no significant difference or relationship between variables. It is often used as a starting point for testing the research hypothesis, and if the results of the study reject the null hypothesis, it suggests that there is a significant difference or relationship between variables.

Alternative Hypothesis

An alternative hypothesis is a statement that assumes there is a significant difference or relationship between variables. It is often used as an alternative to the null hypothesis and is tested against the null hypothesis to determine which statement is more accurate.

Directional Hypothesis

A directional hypothesis is a statement that predicts the direction of the relationship between variables. For example, a researcher might predict that increasing the amount of exercise will result in a decrease in body weight.

Non-directional Hypothesis

A non-directional hypothesis is a statement that predicts the relationship between variables but does not specify the direction. For example, a researcher might predict that there is a relationship between the amount of exercise and body weight, but they do not specify whether increasing or decreasing exercise will affect body weight.

Statistical Hypothesis

A statistical hypothesis is a statement that assumes a particular statistical model or distribution for the data. It is often used in statistical analysis to test the significance of a particular result.

Composite Hypothesis

A composite hypothesis is a statement that assumes more than one condition or outcome. It can be divided into several sub-hypotheses, each of which represents a different possible outcome.

Empirical Hypothesis

An empirical hypothesis is a statement that is based on observed phenomena or data. It is often used in scientific research to develop theories or models that explain the observed phenomena.

Simple Hypothesis

A simple hypothesis is a statement that assumes only one outcome or condition. It is often used in scientific research to test a single variable or factor.

Complex Hypothesis

A complex hypothesis is a statement that assumes multiple outcomes or conditions. It is often used in scientific research to test the effects of multiple variables or factors on a particular outcome.

Applications of Hypothesis

Hypotheses are used in various fields to guide research and make predictions about the outcomes of experiments or observations. Here are some examples of how hypotheses are applied in different fields:

• Science : In scientific research, hypotheses are used to test the validity of theories and models that explain natural phenomena. For example, a hypothesis might be formulated to test the effects of a particular variable on a natural system, such as the effects of climate change on an ecosystem.
• Medicine : In medical research, hypotheses are used to test the effectiveness of treatments and therapies for specific conditions. For example, a hypothesis might be formulated to test the effects of a new drug on a particular disease.
• Psychology : In psychology, hypotheses are used to test theories and models of human behavior and cognition. For example, a hypothesis might be formulated to test the effects of a particular stimulus on the brain or behavior.
• Sociology : In sociology, hypotheses are used to test theories and models of social phenomena, such as the effects of social structures or institutions on human behavior. For example, a hypothesis might be formulated to test the effects of income inequality on crime rates.
• Business : In business research, hypotheses are used to test the validity of theories and models that explain business phenomena, such as consumer behavior or market trends. For example, a hypothesis might be formulated to test the effects of a new marketing campaign on consumer buying behavior.
• Engineering : In engineering, hypotheses are used to test the effectiveness of new technologies or designs. For example, a hypothesis might be formulated to test the efficiency of a new solar panel design.

How to write a Hypothesis

Here are the steps to follow when writing a hypothesis:

Identify the Research Question

The first step is to identify the research question that you want to answer through your study. This question should be clear, specific, and focused. It should be something that can be investigated empirically and that has some relevance or significance in the field.

Conduct a Literature Review

Before writing your hypothesis, it’s essential to conduct a thorough literature review to understand what is already known about the topic. This will help you to identify the research gap and formulate a hypothesis that builds on existing knowledge.

Determine the Variables

The next step is to identify the variables involved in the research question. A variable is any characteristic or factor that can vary or change. There are two types of variables: independent and dependent. The independent variable is the one that is manipulated or changed by the researcher, while the dependent variable is the one that is measured or observed as a result of the independent variable.

Formulate the Hypothesis

Based on the research question and the variables involved, you can now formulate your hypothesis. A hypothesis should be a clear and concise statement that predicts the relationship between the variables. It should be testable through empirical research and based on existing theory or evidence.

Write the Null Hypothesis

The null hypothesis is the opposite of the alternative hypothesis, which is the hypothesis that you are testing. The null hypothesis states that there is no significant difference or relationship between the variables. It is important to write the null hypothesis because it allows you to compare your results with what would be expected by chance.

Refine the Hypothesis

After formulating the hypothesis, it’s important to refine it and make it more precise. This may involve clarifying the variables, specifying the direction of the relationship, or making the hypothesis more testable.

Examples of Hypothesis

Here are a few examples of hypotheses in different fields:

• Psychology : “Increased exposure to violent video games leads to increased aggressive behavior in adolescents.”
• Biology : “Higher levels of carbon dioxide in the atmosphere will lead to increased plant growth.”
• Sociology : “Individuals who grow up in households with higher socioeconomic status will have higher levels of education and income as adults.”
• Education : “Implementing a new teaching method will result in higher student achievement scores.”
• Marketing : “Customers who receive a personalized email will be more likely to make a purchase than those who receive a generic email.”
• Physics : “An increase in temperature will cause an increase in the volume of a gas, assuming all other variables remain constant.”
• Medicine : “Consuming a diet high in saturated fats will increase the risk of developing heart disease.”

Purpose of Hypothesis

The purpose of a hypothesis is to provide a testable explanation for an observed phenomenon or a prediction of a future outcome based on existing knowledge or theories. A hypothesis is an essential part of the scientific method and helps to guide the research process by providing a clear focus for investigation. It enables scientists to design experiments or studies to gather evidence and data that can support or refute the proposed explanation or prediction.

The formulation of a hypothesis is based on existing knowledge, observations, and theories, and it should be specific, testable, and falsifiable. A specific hypothesis helps to define the research question, which is important in the research process as it guides the selection of an appropriate research design and methodology. Testability of the hypothesis means that it can be proven or disproven through empirical data collection and analysis. Falsifiability means that the hypothesis should be formulated in such a way that it can be proven wrong if it is incorrect.

In addition to guiding the research process, the testing of hypotheses can lead to new discoveries and advancements in scientific knowledge. When a hypothesis is supported by the data, it can be used to develop new theories or models to explain the observed phenomenon. When a hypothesis is not supported by the data, it can help to refine existing theories or prompt the development of new hypotheses to explain the phenomenon.

When to use Hypothesis

Here are some common situations in which hypotheses are used:

• In scientific research , hypotheses are used to guide the design of experiments and to help researchers make predictions about the outcomes of those experiments.
• In social science research , hypotheses are used to test theories about human behavior, social relationships, and other phenomena.
• I n business , hypotheses can be used to guide decisions about marketing, product development, and other areas. For example, a hypothesis might be that a new product will sell well in a particular market, and this hypothesis can be tested through market research.

Characteristics of Hypothesis

Here are some common characteristics of a hypothesis:

• Testable : A hypothesis must be able to be tested through observation or experimentation. This means that it must be possible to collect data that will either support or refute the hypothesis.
• Falsifiable : A hypothesis must be able to be proven false if it is not supported by the data. If a hypothesis cannot be falsified, then it is not a scientific hypothesis.
• Clear and concise : A hypothesis should be stated in a clear and concise manner so that it can be easily understood and tested.
• Based on existing knowledge : A hypothesis should be based on existing knowledge and research in the field. It should not be based on personal beliefs or opinions.
• Specific : A hypothesis should be specific in terms of the variables being tested and the predicted outcome. This will help to ensure that the research is focused and well-designed.
• Tentative: A hypothesis is a tentative statement or assumption that requires further testing and evidence to be confirmed or refuted. It is not a final conclusion or assertion.
• Relevant : A hypothesis should be relevant to the research question or problem being studied. It should address a gap in knowledge or provide a new perspective on the issue.

Advantages of Hypothesis

Hypotheses have several advantages in scientific research and experimentation:

• Guides research: A hypothesis provides a clear and specific direction for research. It helps to focus the research question, select appropriate methods and variables, and interpret the results.
• Predictive powe r: A hypothesis makes predictions about the outcome of research, which can be tested through experimentation. This allows researchers to evaluate the validity of the hypothesis and make new discoveries.
• Facilitates communication: A hypothesis provides a common language and framework for scientists to communicate with one another about their research. This helps to facilitate the exchange of ideas and promotes collaboration.
• Efficient use of resources: A hypothesis helps researchers to use their time, resources, and funding efficiently by directing them towards specific research questions and methods that are most likely to yield results.
• Provides a basis for further research: A hypothesis that is supported by data provides a basis for further research and exploration. It can lead to new hypotheses, theories, and discoveries.
• Increases objectivity: A hypothesis can help to increase objectivity in research by providing a clear and specific framework for testing and interpreting results. This can reduce bias and increase the reliability of research findings.

Limitations of Hypothesis

Some Limitations of the Hypothesis are as follows:

• Limited to observable phenomena: Hypotheses are limited to observable phenomena and cannot account for unobservable or intangible factors. This means that some research questions may not be amenable to hypothesis testing.
• May be inaccurate or incomplete: Hypotheses are based on existing knowledge and research, which may be incomplete or inaccurate. This can lead to flawed hypotheses and erroneous conclusions.
• May be biased: Hypotheses may be biased by the researcher’s own beliefs, values, or assumptions. This can lead to selective interpretation of data and a lack of objectivity in research.
• Cannot prove causation: A hypothesis can only show a correlation between variables, but it cannot prove causation. This requires further experimentation and analysis.
• Limited to specific contexts: Hypotheses are limited to specific contexts and may not be generalizable to other situations or populations. This means that results may not be applicable in other contexts or may require further testing.
• May be affected by chance : Hypotheses may be affected by chance or random variation, which can obscure or distort the true relationship between variables.

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I Tried Medium-Rare Chicken. You Should, Too.

It was good you gotta believe me.

The following fact is indisputable: Steak tastes its best when it’s medium rare. The same is true for salmon, tuna, and really, any other cut of quality seafood, which is often served either entirely raw or lightly seared. We have evolved past the outmoded kitchen guidelines that claimed that pork must be cooked to a parched, bone-white opacity, starving the meat of its luxuriant juices. And then there’s duck, which, despite being poultry, tastes most heavenly when it’s crisp on the outside and cherry red in the middle.

When you bundle all of these observations together, you are left with no choice but to conclude that animal protein is most delicious when slightly undone. If you extrapolate this point even further, then surely, undercooked chicken must also be outrageously yummy, and we’ve all been missing out on the epicurean range of America’s favorite dinner plate for generations. It’s a hypothesis worth considering because, if you haven’t noticed, chicken sucks. It’s boring. The amount of attention necessary to inject the faintest whiff of dynamism into the bird has been the bane of chefs for centuries. And if strategic undercooking is the secret to unlocking the protein’s finer qualities, then it must be a noble pursuit. This is the basis of my lifelong fascination with the culinary potential of pink chicken, and why I set out to find a way to sink my teeth into a wad of breast meat cooked to an exquisite medium rare.

I have always been an adventurous eater. I’ve sampled ruby-red horse sashimi in Tokyo, poached duck’s blood in Chongqing, and steamed mantis shrimp—with all of its spindling centipedelike legs intact—in Bangkok. As such, I tend to think Americans are annoying and precious with what they allow into their stomachs. Thankfully, the culture appears to be in the midst of a nutritional reckoning, with countless influencers pushing heterodox eating habits on their platforms. Raw milk is having a moment , so is raw honey , and raw liver . We must also mention the existence of the Instagram account Raw Chicken Experiment , which has garnered over 400,000 Instagram followers, all of whom watch an unnamed man consume raw chicken, day after day, until he gets a “tummy ache.” (Currently, he’s on his 101 st dinner of refrigerator-cold unpasteurized poultry.)

However, it must be reiterated that no food scientist on the planet would endorse the idea of consuming chicken that hasn’t been fully pasteurized. “We risk consuming bacteria which can lead to food poisoning,” said Julia Zumpano, a dietician at the Cleveland Clinic who laid out the assortment of bowel-destroying microbes present in raw chicken, E. coli being the most common. Zumpano, like every other registered dietician, recommends bringing poultry of any variety up to 165 degrees , which is a temperature hot enough to incinerate all of those bacterial agents, guaranteeing a safe digestion. This isn’t a regulatory overreach, either. According to the Centers for Disease Control and Prevention , 1 out of every 25 packages of chicken in the grocery store is contaminated with salmonella, which means that if you are routinely chowing on the rubbery pink of unpasteurized poultry, there is a good chance that you may soon be making several grim treks to the bathroom. Humans have understood this concept for millennia. In American colonial homes, one of the most popular ways to cook chicken was to hang it from a string in front of a fireplace. According to the Greenwich Historical Society, children would often be tasked with spinning the string in front of the hearth, to ensure every part of the bird was fully pasteurized before eating.

But that hasn’t stopped some of the planet’s more intrepid eaters from throwing caution to the wind, and scarfing down raw chicken. After all, if you know where to look, you can find chefs willing to experiment with the dark arts of undercooked poultry. The most famous of these traditions is surely Japan’s notorious torisashi , colloquially known in the Western Hemisphere as chicken sashimi, which is essentially chicken breast that’s either served completely raw or has been put under intense heat for a couple of milliseconds until its left with a charred exterior surrounding a wet, cold, coral-pink interior. Torisashi is hard to find in America, though you can track it down at certain audacious yakitori counters—like the famous Berkeley restaurant Ippuku, which contains a whole gallery on its Yelp page of patrons gawking at its chicken tartare . (“I’ll still give this place three stars even though I got food poisoning,” reads one review. “Other than that it’s a pretty legit joint.”)

Torisashi tends to be more of a regional delicacy in Japan, particularly in the Kyushu city of Miyazaki. The dish has an ardent cult of fans, like 39-year-old New Zealander William Heath, who tells me he was previously married to a woman from Miyazaki. During his trips to the island, Heath estimates he ate torisashi over 200 times, and he rates it as one of his favorite meals.

“It has the texture of sashimi salmon. A meaty yet yielding texture. Most times I’ve eaten it, it’s been with a sear, like a blue steak. Generally with a ponzu sauce, white onions, and wasabi. It’s not slimy like what you’d expect with raw chicken,” he said. “Some places serve it with a raw egg. Imagine that in the Western world! If you can push yourself a little bit beyond what you’ve been taught your whole life, you open up to a whole world of tastes and flavors that are, without being dramatic, awe-inspiring.”

Heath said he was never concerned about the potential health fallout from the dish—particularly when he was eating with other diners who reveled in the sinewy tang of raw breast meat. (“Japanese food is notorious for being stringent to cleanliness,” he said. “And any fast food seen as safe, like McDonald’s or Subway, has the chance to make you ill.”) Of course, Japan’s own Ministry of Health has pushed back on Heath’s assertion that the country has mastered the art of preparing unpasteurized poultry without the risk of personal contamination, to the point of issuing a warning to travelers imploring them to avoid consuming “raw or inadequately cooked chicken” while visiting the island. I’m also not surprised to hear that despite Heath’s fondness for the dish, he’s never attempted to replicate torisashi himself.

“I don’t have the skills, knowledge, or expertise to do it correctly,” he said, comparing torisashi, aptly, to the highly toxic fugu pufferfish that appears on the menu of certain high-end sushi restaurants.

All of this is to say if I wanted to eat undercooked chicken without maxing out my deductible, then torisashi was probably off the table. I needed to think outside the box, which, before long, brought me to the wondrous world of sous vide . The French innovation, in which protein is placed into vacuum-sealed plastic bags and poached in water that has been heated to an uber-precise temperature, is most commonly used to cook red meat. But I had heard that there existed a method to use the machine to bring chicken up to a delectable 140 degrees—the same temperature range for a ruddy medium steak—while still eradicating those pesky colonies of E. coli and salmonella.

The method was popularized by J. Kenji Lopez-Alt, a chef and food writer, and the author of The Food Lab: Better Home Cooking Through Science . In an article he published on Serious Eats , Lopez-Alt argued that pasteurizing chicken is a process that involves both time and heat. Yes, the prescribed 165-degree threshold for poultry will eliminate hostile bacteria in the blink of an eye, but holding the protein at a lower temperature will eventually accomplish the same task over a longer cooking duration. That might be difficult to accomplish on a finicky stovetop, but a sous vide circulator, built to maintain a specific level of heat for hours on end, is perfect for the job.

“At 165 degrees you achieve pasteurization nearly instantly. It’s the bacterial equivalent of shoving a stick of dynamite into an anthill,” wrote Lopez-Alt. “At 136 degrees, on the other hand, it takes a little over an hour for the bacteria to slowly wither to death in the heat.”

Many in the sous vide community have become enthralled by the promise of 140-degree chicken. Cole Wagoner, who works in marketing and frequently shows off the dish on his social feeds , claims that subtemperature poultry is so radically different from the staid blandness of conventional roasted chicken breast that it can almost have a psychotropic effect on a diner’s brain.

“It’s the difference between a medium-rare steak and a well-done steak,” he said. “You cut into it and see an immediate difference. It’s the same flavor, but the amount of natural moisture you get with the sous vide method is profound.”

But Wagoner also mentions that his dish tends to get a polarized response from his dinner guests. Sometimes, after he carves a light-pink chicken breast at the table, his friends and family will whip out their phones and order a circulator for themselves to get in on the revolution. Other folks—like Wagoner’s parents—are so disgusted by the sight that they refuse to even try it.

“I haven’t had many converts,” continued Wagoner. “I haven’t had people say, ‘That looks gross’ and after trying it, they decide they love it.”

After trying the method myself, I can understand where Wagoner is coming from. I arrived at the Slate office kitchen armed with two boneless, skinless chicken breasts, which were subsequently bagged, vacuum-sealed, and dunked in a colleague’s circulator. We set the timer for two hours, at 140 degrees, in accordance with the recipe outlined by Lopez-Alt. I didn’t have high hopes. Poached chicken, in any variety, is never the most visually appealing dish, and once the timer went off, we pulled two grotty, lukewarm hunks of poultry from the depths of the machine. Both of them had turned pallid in their bags, which were stained by the muddy secretion of their juices. Bon appétit ?

To my relief, nothing about the chicken breasts appeared to be viscerally undercooked. Yes, they lacked some of the appetizing wrinkles chefs use to spruce up some of the more tiresome items in their inventory—no grill marks, or cast-iron caramelization, or evidence of a marinade or spice blend—but they didn’t look poisonous, either. My colleague had brought along a kitchen torch, and he sizzled the exterior until the chicken looked less pale and more edible. We got to carving afterward, and the knife passed through the meat with almost no resistance, revealing a few light-pink rings complementing the ruffled whiteness of the protein. My dream had finally come true. Medium-rare chicken was at hand.

Wagoner was right. So was Heath. Undercooked chicken will change everything you believed about cooking poultry. The chicken was unbelievably soft. Almost gelatinous, with the physical consistency of an overnight brisket. It was juicy to the point of being disorienting. Slicing into the breast meat was like puncturing a water balloon—ultra-indulgent and almost sinful, you could peel off splinters of white meat with your fingertips and let them melt in your mouth. The flavor profile didn’t change much, though. This was still definitely chicken, but a heightened, more primal chicken—almost gamey, bearing evidence of once being alive.

But was it good? That’s a question I’ve been struggling to answer. Like most Americans, I have been conditioned to expect a very narrow set of possibilities with my chicken. It is the weeknight protein, a dish that is primarily kept on the menu to cater to fussy eaters, and even at its best—say, a whole roast bird on a perch of root vegetables, golden brown and oozing with rendered butter—the dining experience is pleasantly mild. But at 140 degrees, chicken subverts so many of those comforts that it no longer fits into its domestic reliability. I imagine sitting at a dinner table with my family who are all wide-eyed and zonked-out after experiencing this decadent chicken odyssey—a version of their favorite boring white meat with all of its positive qualities cranked up toward an overripe extreme. We’d be satiated but overwhelmed, and I think that makes sous vide chicken difficult to dish up on a random Tuesday evening.

That said, I was pleased to confirm my theory. Yes, as it turns out, a medium-rare chicken does taste amazing, in perfect lockstep with all the other animals I like to eat. I wrapped up the other breast and packed it away, and started fantasizing about all the ways it could be served. Maybe a medium-rare chicken salad? Or a medium-rare chicken cutlet, ripped out of the sous vide and then breaded and flash-fried? The possibilities were endless. First things first though, I offered a forkful of my experiment to some of my other Slate colleagues, hoping that they, too, would see the light. I was rejected across the board. No surprises there. We may have finally come up with a way to make medium-rare chicken, but it might be much longer before anyone wants to eat it.

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Peer-reviewed

Research Article

A novel kinetic model to demonstrate the independent effects of ATP and ADP/Pi concentrations on sarcomere function

Roles Conceptualization, Data curation, Formal analysis, Funding acquisition, Investigation, Methodology, Project administration, Software, Validation, Visualization, Writing – original draft, Writing – review & editing

Affiliations Department of Biomedical Engineering, University of California, Irvine, Irvine, California, United States of America, UCI Edwards Lifesciences Foundation Cardiovascular Innovation and Research Center (CIRC), University of California, Irvine, Irvine, California, United States of America

Roles Conceptualization, Methodology, Supervision, Validation, Writing – review & editing

Affiliation Department of Physics and Center for Soft Matter Research, New York University, New York, New York, United States of America

Roles Conceptualization, Formal analysis, Funding acquisition, Methodology, Project administration, Resources, Supervision, Validation, Writing – review & editing

* E-mail: [email protected]

Affiliations Department of Biomedical Engineering, University of California, Irvine, Irvine, California, United States of America, UCI Edwards Lifesciences Foundation Cardiovascular Innovation and Research Center (CIRC), University of California, Irvine, Irvine, California, United States of America, Department of Chemical & Biomolecular Engineering, University of California, Irvine, Irvine, California, United States of America, The NSF-Simons Center for Multiscale Cell Fate Research and Sue and Bill Gross Stem Cell Research Center and Center for Complex Biological Systems, University of California, Irvine, Irvine, California, United States of America

• Andrew A. Schmidt, 
• Alexander Y. Grosberg, 
• Anna Grosberg

• Published: August 5, 2024
• https://doi.org/10.1371/journal.pcbi.1012321
• Reader Comments

This is an uncorrected proof.

Understanding muscle contraction mechanisms is a standing challenge, and one of the approaches has been to create models of the sarcomere–the basic contractile unit of striated muscle. While these models have been successful in elucidating many aspects of muscle contraction, they fall short in explaining the energetics of functional phenomena, such as rigor, and in particular, their dependence on the concentrations of the biomolecules involved in the cross-bridge cycle. Our hypothesis posits that the stochastic time delay between ATP adsorption and ADP/Pi release in the cross-bridge cycle necessitates a modeling approach where the rates of these two reaction steps are controlled by two independent parts of the total free energy change of the hydrolysis reaction. To test this hypothesis, we built a two-filament, stochastic-mechanical half-sarcomere model that separates the energetic roles of ATP and ADP/Pi in the cross-bridge cycle’s free energy landscape. Our results clearly demonstrate that there is a nontrivial dependence of the cross-bridge cycle’s kinetics on the independent concentrations of ATP, ADP, and Pi. The simplicity of the proposed model allows for analytical solutions of the more basic systems, which provide novel insight into the dominant mechanisms driving some of the experimentally observed contractile phenomena.

Author summary

Explaining the intricate workings behind muscle contraction remains a fundamental challenge in our field. In this work, we develop a computational model of the sarcomere designed to unravel the basic energetics of sarcomere contraction, and we place major emphasis on the stochastic nature of the reactions in the cross-bridge cycle. The main goal was to illustrate how dynamic processes such as rigor are contingent upon the concentrations of biomolecules governing the kinetics of the cross-bridge cycle. We posited that including the free energy contributions associated with ATP and ADP/Pi as separate reaction steps could unveil previously inaccessible aspects of sarcomere contraction. To test this hypothesis, we constructed a stochastic-mechanical half-sarcomere model whose kinetics explicitly account for the fact that ATP and ADP/Pi interact with myosin at different times in the cross-bridge cycle. Our findings demonstrate a dependence of sarcomere outputs on independent concentrations of ATP, ADP, and Pi, a phenomenon exclusively reproducible with our hypothesized free energy framework. Lastly, the conceptual simplicity of our model enables analytical solutions for elementary systems, affording new insights into the principal drivers governing experimentally observed contractile phenomena.

Citation: Schmidt AA, Grosberg AY, Grosberg A (2024) A novel kinetic model to demonstrate the independent effects of ATP and ADP/Pi concentrations on sarcomere function. PLoS Comput Biol 20(8): e1012321. https://doi.org/10.1371/journal.pcbi.1012321

Editor: Daniel A. Beard, University of Michigan, UNITED STATES OF AMERICA

Received: February 26, 2024; Accepted: July 12, 2024; Published: August 5, 2024

Copyright: © 2024 Schmidt et al. This is an open access article distributed under the terms of the Creative Commons Attribution License , which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

Data Availability: The code for this manuscript is available on github: https://github.com/Cardiovascular-Modeling-Laboratory/SarcomereModel .

Funding: This work was partially supported by NIH T32HL116270 (AS), DoD NDSEG Fellowship (AS), NSF CMMI-2035264 (AG), NSF CMMI-2230503 (AG), NIH R03 EB028605 (AG). The funders did not play a role in the study design, data collection and analysis, decision to publish, nor preparation of the manuscript.

Competing interests: The authors have declared that no competing interests exist.

Introduction

Force generation in striated muscle is regulated by the complex interactions between the actomyosin complex and ATP, ADP, and inorganic phosphate (Pi). Changes in these concentrations can significantly affect muscle contraction, relaxation, and the overall energy balance of contractile cells. Both decreased ATP levels and elevated ADP and Pi levels have been observed in several pathological conditions including heart failure, ischemia, and mitochondrial disorders [ 1 – 5 ]. Despite the importance of investigating the effects of varying ATP, ADP, and Pi concentrations on muscle, the mechanisms that drive the dynamical contractile response are not fully understood.

The generation and maintenance of contractile mechanical stress in striated muscle is performed by sarcomeres black, the basic contractile units of striated muscle. Sarcomeres consist of a three-dimensional lattice of two main types of filaments–thick filaments, which are bound to the center of the sarcomere at the M-line and the ends of the sarcomere at the Z-lines (via titin), and thin filaments, which are bound only at the Z-lines [ 6 , 7 ]. During concentric contraction, the sarcomere shortens as thick filament myosin heads pull thin filaments toward the center of the sarcomere. This pulling force is generated via the cross-bridge cycle, which involves interactions between a single myosin head and a discrete binding site on the actin filament [ 7 , 8 ]. In each cycle, myosin ATPase hydrolyzes one ATP molecule, whose free energy of hydrolysis is partially converted into mechanical work during the power stroke [ 8 , 9 ]. Consequently, ATP availability and the ease of release of its hydrolysis products, ADP and Pi, play integral roles in the possible force generated by the muscle. Therefore, recreating these dynamics of the cross-bridge cycle could be essential for sarcomere models.

For over half a century, a variety of models have been developed to recapitulate the behavior of a sarcomere [ 10 – 27 ]. A review of these models can be found within Niederer et al.’s work [ 28 ] and within the Introduction of Mijailovich et al.’s 2016 MUSICO paper [ 21 ]. Many of the stochastic, spatially explicit models recreate the discrete locations and interactions of the sarcomeric filaments in space (1 to 3 dimensions), allow them to capture the nuances of force generation at a granular level, the propagation of mechanical signals, the heterogeneity in cross-bridge binding and sarcomere lengths, and the internal tension contributions by other compliant components of the sarcomere. However, the kinetic schema of some of these existing models needs to be augmented with rate constants that properly include the free energy contributions of the concentrations of ATP, ADP, and Pi in order to cover a wider variety of experimental conditions [ 14 – 23 ]. A physiologically relevant partitioning of these chemical potentials would also allow for closer examination of the effects of ATP availability, as well as ADP and Pi excess, on sarcomere force generation and maintenance. Conversely, while some probabilistic sarcomere models include the impact of all three molecules in their cross-bridge rate kinetics [ 25 – 27 ], they do not possess the same advantages as stochastic and discrete lattice sarcomere models [ 29 , 30 ].

In this work, we detail the formulation of a spatially explicit, two-filament half-sarcomere model capable of elucidating force generation profiles at varying levels of ATP, ADP, and Pi. Specifically, we employed this model to predict the sarcomeric ATP consumption associated with different levels of contractile force. Thus, we created a novel stochastic-mechanical sarcomere model that tracks discrete node locations and implements a direct dependence of cross-bridge rate kinetics on the concentrations of ATP and its hydrolysis products. The findings of this work yield new insight on the energetics of force generation in muscle tissues.

Methods: Model formulation

We adopt, with small modifications, the two-filament sarcomere model analyzed in previous studies [ 10 , 14 , 16 ]. Our model is a composition of three aspects, the first of which is the half-sarcomere’s geometry. This aspect simplifies the three-dimensional interactions between thick and thin filaments to a one-dimensional system. The second aspect, the mechanics of the half-sarcomere, assumes the sarcomere behaves as a set of linearly elastic (Hookean) springs, and is described by a set of linear equations that combine the geometric constants of the sarcomere with the spring constants of the sarcomere’s physiological components. The final aspect of this model, which underpins the innovation introduced in this paper, is the chemical kinetics, which describe the stochastic chemical transformations through the cross-bridge cycle. Following the majority of previous works, we assume that the elastic equilibrium in the system is achieved comparatively very fast, such that chemical transformations occur essentially between various elastically equilibrated states. To describe the chemical cycle, we adopt the middle ground between the simplest two-state models [ 10 ] and 9 state models [ 30 ], and employ the three-state description, which is widely considered the minimal number of states appropriate for recapitulating a cross-bridge’s biomechanics [ 10 , 14 , 16 , 21 ].

The geometry of the half-sarcomere includes two filaments that are each composed of an array of nodes ( Fig 1 ). Each node on a thin filament ( a n ) represents a discrete actin binding site to which a myosin cross-bridge can bind. Thick filament nodes ( m n ) represent the base of each myosin cross-bridge. In addition to the total number of actin nodes, N a , and the total number myosin nodes, N m , there is a node at the end of each filament: one at the Z-line ( a Z ) and the M-line ( m M ) for the thin and thick filaments respectively. This results in N a + N m + 2 total nodes. Titin was incorporated into this model as a spring element binding the Z-line to the myosin node most distal to the M-line ( Fig 1A , green spring). While Fig 1 is depicted in two dimensions, it is only done so for the clarity of presentation. The forces and displacements in the model are assumed to exist solely along the x-axis parallel to the thick and thin filaments. Elements of physiological spacing were incorporated into this study’s model in order to preserve, at least in part, the properties of the higher order three-dimensional nature of physiological sarcomeres (details in Section A of S1 Text ). While this one-dimensional, two-filament system does not fully capture the three-dimensional helical geometry of a sarcomere in vivo , the simplicity of the system makes it a valuable tool for interrogating the mechanical and chemical dynamics of force generation relevant to this paper.

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(A) Half-sarcomere model showing the geometry of a system containing two actin nodes and two myosin nodes. Cross-bridges are bound, connecting the thick and thin filaments. (B) Schematic showing the three potential mechanical states of a cross-bridge. State 1 shows a cross-bridge unbound from the thin filament. State 2 shows a bound, pre-power stroke cross-bridge in a low force bearing state. State 3 shows a bound, post-power stroke cross-bridge in a high force bearing state. In state 3, the cross-bridge has also undergone a conformational change where the cross-bridge rest length ( b 0 ) has shortened by the length of a power stroke ( d ps ). The model is one dimensional, but this figure illustrates the model in two dimensions for clarity. All forces in this model are assumed to be one-dimensional, parallel to the filaments.

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As cross-bridge binding and/or force generation within the half-sarcomere causes distortions in the spring elements of the model, both K and V vary their components accordingly. At any moment, the mechanical equilibrium of the model, and more specifically the location of each node within the lattice, was calculated from Eq 1 using MATLAB’s internal system of linear equations solver (details in Appendix B).

Shorthand for the biochemical states of the actomyosin complex are displayed in black frames: A-actin, M-myosin, ATP-adenosine triphophate, ADP-adenosine diphosphate, Pi-inorganic phosphate. All elements in each black box are bound. Between state 3 and state 1, ATP binds the actomyosin complex and is hydrolyzed. Rates for the transitions between each state are labeled such that k ij represents the transition rate from state i to j . Association of ATP to the actomyosin complex and the dissociation of ADP and Pi from the actomyosin complex are indicated at the appropriate transition.

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Shorthand of the biochemical states of the actomyosin complex are framed in black (key in Fig 2 caption). Transition states are denoted by dashed lines and dashed frames. Δ G hyd = Δ G T assoc. + Δ G T hydr. + Δ G D,Pi rel. . Free energy of association of ATP is Δ G T assoc . . Free energy of ATP hydrolysis is Δ G T hydr. . Free energy of of ADP and Pi release from actomyosin is Δ G D,Pi rel. . G ADP,Pi = k B T ln([ADP][Pi]). G ATP = k B T ln([ATP]). Note: The free energies of each state depicted in this free energy landscape assume there is no cross-bridge deformation, and therefore do not include the elastic potential energy contributions of such deformations. The complete free energies of each state, including elastic potential energies, are fully defined in Eqs 2 – 5 .

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A reference energy of 0 was set for the free energy of the state 1 ( Eq 2 ). Traveling through each cycle requires the addition of an ATP from the environment, thus we defined state 1’ as the energy baseline of the following new cross-bridge cycle ( Eq 5 ). All parameters were taken from literature (details in Section D of S1 Text ), and then fine-tuned to match the physiological duty ratio [ 31 ]. Fine tuning of parameters may need to be adjusted within their pre-determined ranges depending on the specific geometry of the system. For a one-myosin system, all variables and parameters are defined in Table 1 . The rate constants along with their equilibrium ratios were defined as follows, using the notations from Table 1 :

Variables and the corresponding units or values used in the sarcomere model. Values were selected after a parameter exploration was performed on a range of values pulled from literature from both models and experiments (details in Section D of S1 Text ).

https://doi.org/10.1371/journal.pcbi.1012321.t001

The solution implementation in MATLAB is discussed in detail in Sections B and C of S1 Text . For all simulations in this investigation, the sarcomere was allowed to spontaneously contract (i.e. no assigned velocity of shortening), with full calcium activation of all binding sites (actin nodes) along the thin filament.

The model implementation was verified against the inherent physics of the system. For example, we compared the energy input into the system via ATP hydrolysis to the total elastic potential energy of the springs in the system. Model outputs were compared to those reported in experimental literature, such as ranges of values for ATP consumption, peak force of a single myosin, and force per myosin in larger systems. Independently, reports of rigor concentrations of ATP were compared to predictions made by this model.

Before investigating the impact of the new free energy schema on sarcomere behavior across a range of [ATP] and [ADP][Pi] concentrations, we first validated the model by simulating a single myosin system at the standard normal concentrations for ATP, ADP, and Pi: 5 mM, 0.03 mM, and 3 mM respectively [ 13 , 29 , 52 , 53 ]. This simulation revealed a peak force of 3 pN per myosin. This value is consistent with a previous literature range of ∼1–7 pN [ 32 , 54 – 57 ]. Next, a half-sarcomere consisting of 16 myosin and 24 actin nodes was simulated ( Fig 4A ) with the same parameters as a single myosin system ( Table 1 ). The average sarcomeric force output was 2.1 pN (dashed line Fig 4A ), resulting in a time-averaged force per myosin of 0.13 pN. The estimation method described previously for organ-scale contraction [ 58 ] was applied to data from other experiments, including muscular thin films (tissue-scale) and traction force microscopy (cell-scale) [ 59 – 63 ], which resulted in the range of forces per myosin in different systems to be 0.04–1 pN. Thus, we conclude that our values of time-averaged force/myosin are within physiologically expected ranges.

Single half-sarcomere consisting of 16 myosin and 24 actin nodes under (A-B) normal ([ATP] = 5 mM) and (C-D) reduced ([ATP] = 0.5μM) ATP conditions. (A,C) Force output profile denoted by black circles. Average force denoted by the dashed line. Force averaged over a sliding window of (A) 12 ms and (C) 14 ms denoted by blue lines with blue diamonds. (B,D) ATP consumption rate (molecules/s) denoted by black circles. ATP consumption rate averaged over a sliding window of 50 ms denoted by green diamonds.

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To validate the order of magnitude of ATPase activity, ATP consumption rate per myosin was calculated by approximating the density of myosin heads per muscle tissue volume (0.48–1.2 × 10 17 myosin/cm 3 ) from literature estimates of myosin concentration in muscle [ 64 – 67 ]. Based on experimental data from isometrically and concentrically contracting muscle, an ATP consumption estimate would be on the order of 1–120ATP/s per myosin [ 41 , 65 – 76 ]. If one takes into account the proportion of myosin hypothesized to actually be participating in contraction [ 77 ], an estimate would yield a range of 2–240ATP/s per myosin (further details in Section E of S1 Text ). In the model, ATPase activity was then quantified by tracking the number of myosin transitions (per unit time) from state 3 to state 1–the transition that involves the hydrolysis of one ATP molecule ( Fig 4B ). To avoid biases from the initialization of the contraction simulation, plateau ATP consumption rates were calculated as the mean rate after the rate first exceeds 98% of the maximum consumption rate. The plateau ATP consumption rate of the single 16-myosin sarcomere system was 1400ATP/s ( Fig 4B ) or about 88ATP/s per myosin, matching our physiological estimates.

Having validated the model, we next demonstrated that there was a significant change to the behavior of the sarcomere when the concentration of [ATP] was changed to 0.5 μM ( Fig 4C and 4D ), while concentrations of [ADP] and [Pi] were maintained at those found under standard conditions. As can be seen from the force plot ( Fig 4C ), the sarcomere exhibited rigor like behavior with slow “ratcheting”–characterized by repeated cycles of brief contractile force increases followed by periods of force stagnancy due to lack of ATP. The effect of reduced [ATP] also manifests itself in the ATP consumption rate ( Fig 4D ), which is significantly reduced compared to that of the system under standard conditions.

To further explore this, we considered situations with varying values of concentrations of [ATP] and [ADP][Pi]. If all transition rates were to be assumed dependent only upon the ratio [ATP]/[ADP][Pi], all fluctuations in the sarcomere, as well as the average times a myosin head spends in states 1, 2, and 3, would also solely depend on only the ratio [ATP]/[ADP][Pi]. Fig 5A demonstrates how very different values of [ATP] and [ADP][Pi] can have ratios that are the same (equivalent ratios displayed in the same color), resulting in the diagonal symmetry. This point is illustrated in Fig 5B , where we show what the ATP consumption for a single myosin system would look like if state transitions were dependent upon only the ratio [ATP]/[ADP][Pi]. Such a system would have state transition rates that are equivalent as long as the ratio [ATP]/[ADP][Pi] is the same.

Comparison of free energy schema in terms of ATP consumption (A) Ratios of [ATP] to [ADP][Pi]. (B) Plateau ATP consumption rate for a one-myosin system where attachment of ATP and detachment of ADP and Pi effectively happen simultaneously, allowing for no time delay between these events. For such a model, the rate kinetics, and thus ATP consumption, depend only the ratio of [ATP]/[ADP][Pi]. (C) Our model’s plateau ATP consumption rate simulation results for a one-myosin system. Note the plot’s asymmetry compared to (B) and the proximity of the standard physiological conditions (bold red lines) to the crossover between regimes. One-myosin simulation results of (D) Duty ratio and (E) Average force for various ([ATP], [ADP][Pi]) combinations. (F) Changes in duty ratio (black), average force output (purple), and plateau ATP consumption rate (blue) at standard physiological [ADP][Pi] and varying [ATP] concentrations. (B-F) Standard physiological conditions are denoted by bold red lines ([ATP] = 5 mM, [ADP][Pi] = 0.09 mM 2 ). Dashed white/gray lines denote the the [ATP] concentration associated with the onset of rigor ([ATP] = 0.5 mM) [ 75 , 79 ]. (C-F) n = 10.

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The ability of our model to consider separately how [ATP], [ADP], and [Pi] affect the transition rates allowed us to interrogate imbalances in these molecules (Eqs 2 – 14 ). Our model demonstrated that, for a single myosin simulation across varying ([ATP], [ADP][Pi]) combinations, there is in fact an asymmetry in ATP consumption that directly results from the appropriate allocation of the overall free energy change of the cross-bridge cycle ( Fig 5C ). Importantly, the physiological concentration of [ATP] falls within the range of the observed transition region roughly from 10mM to 5 μM (yellow to blue, Fig 5C ). This change in regime is also visible in one myosin system simulation outputs for duty ratio (the fraction of time a myosin head spends in state 3 [ 31 , 48 , 78 ]) and average force ( Fig 5D–5F ).

Furthermore, the ATP concentration at which the system begins to exhibit rigor-like characteristics (dashed white/gray line, Fig 5C–5F ), indicated by a rise in duty ratio, is consistent with the one experimentally observed with the onset of rigor ([ATP] ≤ 0.5 mM) [ 75 , 79 ]. Notably, the free energy schema utilized by this model was constructed completely independent of any experimental results on rigor-inducing concentrations of [ATP]. Therefore, the alignment between our model’s predictions and experimental values acts as an independent validation of the proposed free energy schema.

Next, we examined duty ratio, average force, and plateau ATP consumption rate for a simulation of a 16-myosin half-sarcomere model with the same parameters as a single myosin system ( Table 1 ) across varying ([ATP], [ADP][Pi]) combinations and qualitatively observed similar asymmetry in duty ratio, average force, and plateau ATP consumption rates ( Fig 6A–6C ). For example, in the region with lower than normal [ATP], and low to normal [ADP][Pi], multi-myosin half-sarcomere systems will ratchet up the thin filament to a greater degree of shortening (Figs 4C and 6A–6C ). This ratcheting behavior is what is expected for muscle that can contract but not relax. While in other regions, such as physiological-adjacent conditions, the system’s force output fluctuates more naturally (Figs 4A and 6A–6C ). In examining the regime change associated with the onset of rigor at normal [ADP][Pi] (0.09 mM 2 ), we noted a shift from where the transition is reported physiologically by one order of magnitude ( Fig 6D–6E ). We hypothesized that this is likely caused by the more complex internal tensions of a multi-myosin system shifting the duty ratio to lower than normal ( Fig 6A and 6D ). The first step in testing this hypothesis was to evaluate whether internal tensions can change expected sarcomere deformations. Indeed, the effective sliding distance (ESD) of the one-myosin system simulation is actually less than the prescribed d ps by approximately 1 nm. Moreover, when comparing one-myosin and multi-myosin systems, the system with more myosin has a larger percentage of the its total elastic potential energy stored within the sarcomere’s internal spring elements (i.e. actin, myosin, titin, bound cross-bridges) as opposed to to the external spring element (i.e. the substrate sarcomere is contracting against), also impacting the ESD ( Fig 6F ).

Results for a (n = 10) showing (A,D) Duty ratio, (B,E) Average force, and (C, D) Plateau ATP consumption rate. Average force and plateau ATP consumption rate are reported as per whole sarcomere. (D) Duty ratio and ATP consumption rate and (E) Average force at standard physiological [ADP][Pi] and varying [ATP] concentrations. (A-E) Standard physiological concentrations for [ATP] and [ADP][Pi] conditions are denoted by the bold red lines ([ATP] = 5 mM, [ADP][Pi] = 0.09 mM 2 ). Dashed white/gray lines represent literature values of [ATP] concentrations where the onset of rigor has been observed experimentally ([ATP] = 0.5 mM) [ 75 , 79 ]. (F) Percent of total elastic potential energy in the spring elements internal to the sarcomere (i.e. actin, myosin, titin, and cross-bridges) and external to the sarcomere (i.e. external spring).

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The analytical outputs for a single myosin system ( Fig 7A–7C ) align with the simulation results ( Fig 5C–5E ) if the effective sliding distance, ESD = 6 nm. To accomplish this, we replaced the cross-bridge displacement term ( Table 1 ) in Eqs 9 , 11 – 13 and 16 – 21 as follows: ( x m − x a − b 0 ) = −ESD = − 6 nm. In contrast, the match is not as close when the ESD is assumed to be the same as prescribed: ESD = d ps = 7 nm ( Fig 7D ). This implies that if our hypothesis is correct, smaller ESDs would lead to analytical solutions that mimic the 16-myosin simulation ( Fig 6A ), which was indeed observed at ESD ≈ 3 nm ( Fig 7E ).

(A) Analytically calculated duty ratio ( Eq 16 ) where the power stroke’s effective sliding distance (ESD) = 6 nm. (B) Analytically calculated average force ( Eq 17 ) where ESD = 6 nm. (C) Analytically calculated ATP consumption rate ( Eq 18 ) where ESD = 6 nm. (D) Analytically calculated duty ratio where ESD = d ps = 7 nm, where 7 nm is the power stroke distance against no resistance. This causes an upward and rightward shift in the plot. (E) Analytically calculated duty ratio where ESD = 3 nm. This causes a downward and leftward shift in the plot. (F) Analytically calculated duty ratio where ESD = 6 nm and k ATP ,0 = 7 × 10 −5 s −1 . This causes a rightward shift in the plot. (A-I) Standard physiological concentrations for [ATP] and [ADP][Pi] conditions are denoted by the bold red lines ([ATP] = 5 mM, [ADP][Pi] = 0.09 mM 2 ). Dashed white/gray lines denote the [ATP] concentration associated with the onset of rigor ([ATP] = 0.5 mM) [ 75 , 79 ]. Effect of reducing k ATP ,0 on one-myosin analytical and 16-myosin simulation (n = 10) values for (G) Duty ratio, (H) Average force, and (I) Plateau ATP consumption rate at standard physiological [ADP][Pi] or increased [ADP][Pi] and varying [ATP] concentrations. Reducing k ATP ,0 causes a rightward shift in all of the plots. Increasing environmental [ADP][Pi] by a factor of 10 3 alters 16 myosin system behavior. Average force and plateau ATP consumption rate are reported as per whole sarcomere. (H, inset) The one-myosin analytical system’s predictions of average force compared to muscle strip data (red circles) adapted from White [ 79 ]. Data are normalized to force under complete rigor, at [ATP] = 0.1 μM for the analytical system and [ATP] = 0mM for experimental data.

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Although the effective sliding distance cannot be prescribed in a simulation, the phenomena of internal tensions can be manipulated by extending the duration each myosin head remains attached to actin, controlled by adjusting the k ATP ,0 parameter in Eqs 13 and 14 while holding all other parameters as in Table 1 . Indeed, by reducing k ATP ,0 from 10 −2 ] s −1 to 7 × 10 −5 s −1 , we observe a rightward shift in the model’s outputs ( Fig 7F–7I ). A 16-myosin sarcomere simulated with this shift enables the system to exhibit the same average duty ratio among its cross-bridges at physiological [ATP], [ADP][Pi] as a one-myosin system with the original k ATP,0 ( Fig 7G ). Excitingly, this realignment towards a physiological duty ratio is concurrent with the shift in the 16-myosin system’s rigor behavior, matching physiological expectations ( Fig 7G–7I ). A feature exclusive to a shorter ESD system, e.g. the 16-myosin system, is its sensitivity to [ADP][Pi] changes near physiological conditions. For example, if environmental [ADP][Pi] is increased by a factor of 10 3 , there are significant shifts in physiological and rigor associated force outputs and ATP consumption (dashed green line, Fig 7H and 7I ).

A sarcomere is the fundamental unit of muscle contraction, and modeling its behavior can provide insight into the mechanisms of muscle function. The model in this study bridges the gap between stochastic-mechanical sarcomere models and a novel cross-bridge cycle kinetic schema that considers [ATP], [ADP], and [Pi] in their relevant state transitions, conferring it the advantages of both types of models ( Fig 3 ). The rate constant definitions in our model were derived with few underlying assumptions, enabling them to be simpler and more direct than in previous models (Eqs 15 , 6 and 12 ), while also effectively capturing how changes in free energy–and consequently kinetic rates–arise from independent perturbations in either [ATP] or [ADP][Pi].

Utilizing our new kinetic schema, we showed that, for each geometry, as long as the duty ratio remained physiological at standard [ATP], [ADP][Pi] levels, it was possible to predict the concentration at which the onset of rigor is expected (Figs 5C–5F , 7A–7C, 7G–7I ), with the analytical solution agreeing remarkably with experimental data [ 79 ] ( Fig 7H , inset). This approach allowed us to observe how shifts in sarcomere behavior could arise from changes in internal mechanics or kinetic adjustments, demonstrated in this study by varying the effective sliding distance or adjusting the parameter k ATP ,0 . The results suggest that k ATP ,0 is a key parameter in fine-tuning the model’s accuracy and relevance in more complex systems ( Fig 7G–7I ). The backward transition that k ATP ,0 governs may be even lower for 3D systems, whose internal tensions are expected to be even more intricate than our 16-myosin system’s ( Fig 6F and contrasting Fig 6D and 6E with Fig 7G–7I ). Modifying k ATP ,0 to account for internal tensions is consistent with theoretical considerations of cross-bridge cycling under increased internal tensions [ 80 , 81 ] and experimental insights of myosin binding kinetics’ role in muscle function [ 82 ], and is thus a valid means of maintaining model fidelity when investigating increasingly complex muscular systems. Therefore, by simple tuning of the duty ratio, a key feature of this model, the kinetic schema can be applied to studies of different myosin classes and isoforms, conditions characterized by altered muscle energetics, and muscle adaptation to energy stress.

Consistent with the results of our model (Figs 4 – 7 ), it is well documented that a sarcomere’s environmental conditions can lead to increased myosin binding to actin (e.g. low-ATP concentrations, high-ADP concentrations, or rigor conditions) or decreased myosin binding to actin (e.g. high-Pi concentration or myosin inhibitors) [ 83 – 86 ], as is evident in many pathologies [ 1 – 5 ]. Beyond the scale of myosin binding, changes in concentrations of any one of these metabolites has shown varied effects on force metrics of muscle fibers [ 87 – 89 ], in alignment with our results on force output ( Fig 7B and 7H ). Even in non-pathological states, such as during muscle fatigue, [ADP] and [Pi] concentrations can increase significantly (20 to 300-fold and 6 to 10-fold respectively–an increase up to the order of 10 3 in [ADP][Pi] [ 52 , 90 – 94 ]), leading to changes in force output and energetics, which our model can explore ( Fig 7H and 7I ). Taken together, the existence of these conditions where the concentrations of these metabolites move independent from one another necessitates contraction models that are capable of interrogating the effects of imbalances in the [ATP]/[ADP][Pi] ratio based on individual concentrations.

As this model was intentionally made to be simple, it does not include additional intermediate states of the cross-bridge cycle or more complex characteristics of a real sarcomere. For example, according to recent discoveries, the super-relaxed (SRX) state of myosin is important due to its potential role in optimizing sarcomeric energy utilization [ 77 , 95 , 96 ]. Thus, SRX states may need to be included in future iterations of the model as it has been suggested that altered concentrations of environmental [ADP] may cause strain-mediated destabilization of the SRX population in sarcomeres [ 77 , 95 ]. While the phenomenological implementation of this SRX state has been previously modeled [ 24 , 97 ], to include this additional cross-bridge state mechanistically within our model would require more experimental data.

The force traces analyzed in this study observe a half-sarcomere system as it contracts against an external spring, so there is no prescribed velocity of shortening or isotonic shortening ( Fig 4A ). This implementation closely mimics the experimental conditions under which cellular and tissue muscle mechanics are studied, such as in traction force microscopy and muscular thin films [ 59 – 63 ]. Therefore, the model can be paired with “heart-chip” experiments that explore the effect of hypoxia [ 98 ] or other altered [ATP], [ADP], [Pi] conditions to predict reductions in contractility. Similarly, in the future the kinetic schema developed here can be hybridized with prescribed velocity contraction models to explore how power density of the muscle changes with varying [ATP] [ 99 ].

This simple model adequately captured the geometries, mechanics, and kinetics necessary for investigating the chemical and mechanical outputs of sarcomeric force generation while also providing the flexibility to interrogate sarcomeric response to alterations in mechanical and kinetic parameters. The novel approach to the allocation of free energies within this model enabled evaluation of sarcomeric outputs in response to changes in [ATP], [ADP], and [Pi] concentrations that would be otherwise misrepresented. The associated analytical solution for the one-myosin system is a powerful tool to explore how different parameters influence stochastic-mechanical behavior based on how these parameters affect the analytical system. The gained insights not only validate the utility of our model but also establish a solid foundation for future experimental explorations aimed at targeting muscular disorders at a molecular level. Since our model possesses geometric and mechanical elements generally consistent with those of previous models, our novel kinetic schema is an easily integrated augmentation that will lead to this work’s increased relevance as a tool to interrogate energy utilization and force generation of sarcomeres under a variety of ATP, ADP, and Pi environmental conditions.

Supporting information

S1 text. this file contains s1 text sections a–e with details on model derivation, implementation, and parameter exploration, and further citations supporting parameter and method selection..

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Antibodies From Long COVID Patients Provide Clues to Autoimmunity Hypothesis

BY ISABELLA BACKMAN August 5, 2024

Promising new research supports that autoimmunity—in which the immune system targets its own body—may contribute to Long COVID symptoms in some patients.

As covered previously in this blog, researchers have several hypotheses to explain what causes Long COVID, including lingering viral remnants, the reactivation of latent viruses, tissue damage, and autoimmunity.

Now, in a recent study , when researchers gave healthy mice antibodies from patients with Long COVID, some of the animals began showing Long COVID symptoms—specifically heightened pain sensitivity and dizziness. It is among the first studies to offer enticing evidence for the autoimmunity hypothesis. The research was led by Akiko Iwasaki, PhD , Sterling Professor of Immunobiology at Yale School of Medicine (YSM).

“We believe this is a big step forward in trying to understand and provide treatment to patients with this subset of Long COVID,” Iwasaki said.

Iwasaki zeroed in on autoimmunity in this study for several reasons. First, Long COVID’s persistent nature suggested that a chronic triggering of the immune system might be at play. Second, women between ages 30 and 50, who are most susceptible to autoimmune diseases, are also at a heightened risk for Long COVID. Finally, some of Iwasaki’s previous research had detected heightened levels of antibodies in people infected with SARS-CoV-2.

Mice given antibodies show signs of Long COVID symptoms

Iwasaki’s team isolated antibodies from blood samples obtained from the Mount Sinai-Yale Long COVID study . They transferred these antibodies into mice and then conducted multiple experiments designed to look for changes in behavior that may indicate the presence of specific symptoms. For many of these experiments, mice that received antibodies [the experimental group] behaved no differently than mice that had not [the control group].

However, a few experiments revealed striking changes in the behavior of the experimental mice. These included:

• Pain sensitivity test: Some experimental mice were quicker to react after being placed on a heated plate.
• Coordination and balance test: Some experimental mice struggled to balance on a rotarod (rotating rod) compared to control mice.
• Grip strength test: Some of the experimental mice applied less force with their paws.

Among the mice that showed behavioral changes, the researchers identified which patients their antibodies came from and what symptoms they had experienced. Interestingly, of the mice that showed heightened pain, 85% received antibodies from patients that reported pain as one of their Long COVID symptoms. Additionally, 89% of mice that had demonstrated loss of balance and coordination on the rotarod test had received antibodies from patients who reported dizziness. Furthermore, 91% of mice that showed reduced strength and muscle weakness received antibodies from patients who reported headache and 55% from patients who reported tinnitus. More research is needed to better understand this correlation.

The autoimmunity hypothesis has recently been further supported by a research group in the Netherlands led by Jeroen den Dunnen, DRS , associate professor at Amsterdam University Medical Center, which also found a link between patients’ Long COVID antibodies and corresponding symptoms in mice.

Treatments for autoimmunity may help some Long COVID patients

Diagnosing and treating Long COVID requires doctors to understand what causes the disease. The new study suggests that treatments targeting autoimmunity, such as B cell depletion therapy or plasmapheresis, might alleviate symptoms in some patients by removing the disease-causing antibodies.

Intravenous immunoglobulin (IVIg) is another therapy used for treating autoimmune diseases like lupus in which patients receive antibodies from healthy donors. While its exact mechanism is still unclear, the treatment can help modulate the immune system and reduce inflammation. Could this treatment help cases of Long COVID that are caused by autoimmunity?

A 2024 study led by Lindsey McAlpine, MD , instructor at YSM and first author, and Serena Spudich, MD , Gilbert H. Glaser Professor of Neurology at YSM and principal investigator, found that IVIg might help improve small fiber neuropathy—a condition associated with numbness or painful sensations in the hands and feet—caused by Long COVID. Iwasaki is hopeful that future clinical trials might reveal the benefits of this treatment in helping some of the other painful symptoms of the diseases.

Other drugs are also in the pipeline, such as FcRn inhibitors. FcRn is a receptor that binds to antibodies and recycles them. Blocking this receptor could help bring down levels of circulating antibodies in the blood. An FcRn receptor was recently approved by the FDA for treating myasthenia gravis, another kind of autoimmune disease.

The study could also help researchers create diagnostic tools for evaluating which patients have Long COVID induced by autoimmunity so that doctors can identify who is most likely to benefit from treatments such as these.

Iwasaki plans to continue researching why and how autoantibodies might cause Long COVID, as well as conduct randomized clinical trials on promising treatments. She is also conducting similar antibody transfer studies in other post-acute infection syndromes, such as myalgic encephalomyelitis/chronic fatigue syndrome (ME/CFS).

In the meantime, she is excited about her team’s promising results. “Seeing this one-to-one correlation of antibodies that cause pain from patients who reported pain is really gratifying to me as it suggests a causal link,” she says. “It’s a first step, but I think it’s a big one.”

Isabella Backman is associate editor and writer at Yale School of Medicine.

The last word by Lisa Sanders, MD:

I am very excited by this research, which suggests that at least some of the symptoms of Long COVID are driven by autoimmunity. If so, then this suggests that there may be a way to test for some versions of Long COVID. And if we could identify the patients who have an autoimmune-driven disease, we have treatments to try that have been used with success in other autoimmune diseases. Many of the autoimmune diseases are treated with medications that suppress the immune system. These are powerful medicines that can leave an individual at risk for infection, so they must be thoughtfully applied to patients with evidence of immune system involvement.

I feel as though every blog post here ends with the possibility of better testing and better treatment, but what makes this different is that it points in a very specific direction and leads to the kind of specific questions that help get to useful answers. Which antibodies are involved? Which cells? And finally, can we develop treatments that are specific to those antibodies or to their targets? These are exciting questions, which will, I hope, lead to useful answers.

Read other installments of Long COVID Dispatches here .

If you’d like to share your experience with Long COVID for possible use in a future post (under a pseudonym), write to us at: [email protected]

Information provided in Yale Medicine content is for general informational purposes only. It should never be used as a substitute for medical advice from your doctor or other qualified clinician. Always seek the individual advice of your health care provider for any questions you have regarding a medical condition.

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Physical Review A

Covering atomic, molecular, and optical physics and quantum science.

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Error estimation of different schemes to measure spin-squeezing inequalities

Jan lennart bönsel, satoya imai, ye-chao liu, and otfried gühne, phys. rev. a 110 , 022410 – published 7 august 2024.

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• INTRODUCTION
• PRELIMINARIES
• THREE WAYS TO MEASURE SPIN-SQUEEZING…
• STATISTICAL ANALYSIS
• ACKNOWLEDGMENTS

How can we analyze quantum correlations in large and noisy systems without quantum state tomography? An established method is to measure total angular momenta and employ the so-called spin-squeezing inequalities based on their expectations and variances. This allows detection of metrologically useful entanglement, but efficient strategies for estimating such nonlinear quantities have yet to be determined. In this paper we show that spin-squeezing inequalities can not only be evaluated by measurements of the total angular momentum but also by two-qubit correlations, either involving all pair correlations or randomly chosen pair correlations. Then we analyze the estimation errors of our approaches in terms of a hypothesis test. For this purpose, we discuss how error bounds can be derived for nonlinear estimators with the help of their variances, characterizing the probability of falsely detecting a separable state as entangled. We focus on the spin-squeezing inequalities in multiqubit systems. Our methods, however, can also be applied to spin-squeezing inequalities for qudits or for the statistical treatment of other nonlinear parameters of quantum states.

• Received 16 January 2024
• Accepted 17 July 2024

DOI: https://doi.org/10.1103/PhysRevA.110.022410

©2024 American Physical Society

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• 1 Naturwissenschaftlich-Technische Fakultät, Universität Siegen , Walter-Flex-Straße 3, 57068 Siegen, Germany
• 2 QSTAR , INO-CNR , and LENS , Largo Enrico Fermi 2, 50125 Firenze, Italy

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Vol. 110, Iss. 2 — August 2024

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Visualization of the singlet state | Ψ − 〉 (red) and the Dicke state | D N , N / 2 〉 (blue) on the collective Bloch sphere [ 40 ]. Singlet states are characterized by vanishing mean spin 〈 J ⃗ 〉 = 0 and variances. Hence, the singlet state | Ψ − 〉 corresponds to the red dot at the origin. The Dicke state | D N , N / 2 〉 is also at the origin, though it has a nonzero variance in the x − y plane that is shown by the blue shaded area.

Upper bound of the p value. The plot shows an exemplary probability density function f ξ ̃ of the estimator for a separable state with spin-squeezing parameter ξ = ξ s . ξ s denotes the extremal value that can be achieved by separable states. To observe an outcome ξ 1 , the estimator has to deviate at least by t = ξ 1 − ξ s from its mean. The probability P ( ξ ̃ − ξ ≥ t ) for this to happen corresponds to the red area.

Measurement scheme for the estimators 〈 J α 2 〉 ̃ TS and ( Δ J α ) 2 ̃ TS . In each repetition k , the total spin of the system is measured. In an ion trap, this can be done by resonance fluorescence [ 34 ], which also gives access to the spin of the individual qubits. The figure includes an image of trapped 171 Yb + ions, which is reprinted from [ 34 ].

Measurement pattern for (a)  〈 J α 2 〉 ̃ AP in Eq. ( 20 ) as well as ( Δ J α ) 2 ̃ AP in Eq. ( 21 ) and (b)  〈 J α 〉 2 ̃ AP in Eq. ( 23 ). In pattern (a) all N ( N − 1 ) distinct pairs of qubits are measured K AP times. In contrast, in pattern (b) all N 2 pairs are measured, with each qubit observed only in K AP 2 of the experimental runs to ensure statistical independence. The approach AP1 relies only on the measurement pattern (a), whereas for AP2 both the patterns (a) and (b) are used.

Measurement pattern for (a)  〈 J α 2 〉 ̃ RP in Eq. ( 25 ) and ( Δ J α ) 2 ̃ RP in Eq. ( 27 ) and for (b)  〈 J α 〉 2 ̃ RP in Eq. ( 29 ). In pattern (a), L RP random pair correlations are measured K RP -times each. Pattern (b) in turn uses also L RP random pairs ( i , j ) , but with the possibility that i = j . In K RP 2 of the repetitions qubit i is measured, whereas in the other repetitions qubit j is observed. The scheme RP1 is only based on the pattern (a), whereas RP2 relies on both patterns (a) and (b).

Probability distribution of the estimator ( ξ ̃ c ) TS . The simulation has been performed for the 10-qubit Dicke state | D 10 , 5 〉 defined in Eq. ( 9 ) with K TS = 7400 . The histogram contains 99 bins, but due to the small bin size of 0.02 they are not well resolved.

Probability distribution of the estimator ( ξ ̃ c ) RP1 . The simulation has been performed for the 10-qubit Dicke state | D 10 , 5 〉 . L RP1 = 7400 random pairs have been chosen with K RP1 = 1 repetitions. The histogram consists of 99 bins with a size of 0.2.

Variances of the estimators ( ξ ̃ c ) TS ,   ( ξ ̃ c ) AP1 ,   ( ξ ̃ c ) AP2 ,   ( ξ ̃ c ) RP1 , and ( ξ ̃ c ) RP2 for the Dicke state of N = 10 qubits | D 10 , 5 〉 mixed with depolarization noise, i.e., ρ = p | D 10 , 5 〉 〈 D 10 , 5 | + ( 1 − p ) 1 / 2 N . The variances are obtained for K TS = 7400 ,   K AP1 = 82 ,   K AP2 = 60 ,   L RP1 = 7400 with K RP1 = 1 and L RP2 = 2775 with K RP2 = 2 .

Number of state preparations necessary to verify a violation of Eq. ( 6c ) by t = 0.1 × N 2 with a significance level of γ = 0.95 .

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Trump Campaign Criticizes Walz for State Law Providing Tampons in Schools

The law, which was passed in Minnesota last year, includes language requiring menstrual products to be available in bathrooms of all schools for grades 4 to 12 as a way to accommodate transgender students.

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By Chris Cameron

• Published Aug. 6, 2024 Updated Aug. 7, 2024, 12:21 p.m. ET

As part of their effort to portray Tim Walz, the new Democratic vice-presidential candidate, as a far-left liberal, the Trump campaign attacked the Minnesota governor on Tuesday for signing a bill last year that provides access to menstrual products for transgender students.

At issue is broadly inclusive language in the law, which states that products like pads, tampons and other products used for menstruation “must be available to all menstruating students in restrooms regularly used by students in grades 4 to 12.” Republican state lawmakers in Minnesota had tried — and failed — to amend that bill so that it would apply only to “female restrooms,” though some Republicans went on to vote for the final version of the bill .

Karoline Leavitt, a spokeswoman for the Trump campaign, said in an interview on Tuesday on Fox News that the law, among other policies seen as supportive of transgender rights, was “a threat to women’s health.”

“As a woman, I think there is no greater threat to our health than leaders who support gender-transition surgeries for young minors , who support putting tampons in men’s bathrooms in public schools,” Ms. Leavitt said. “Those are radical policies that Tim Walz supports. He actually signed a bill to do that.”

State Representative Sandra Feist , a Democrat and the chief author of the bill, said in an interview that it was important for her and the student activists who pushed for the change that transgender students were able to access menstrual products without having to ask for them.

“I actually received emails,” Ms. Feist said. “From trans students, parents, teachers, librarians, custodians from across the country, talking about how they were — or that they knew — trans students who faced these barriers and needed these products, and how much it meant to them that they would have that access, and also that we were standing up for them.”

Mr. Walz made significant efforts to protect the rights of L.G.B.T.Q. people in Minnesota as governor, and was an early supporter of gay rights going as far back as his time as a high school teacher in the 1990s. Mr. Walz signed a bill last year designating Minnesota as a legal refuge for transgender people.

Chris Cameron covers politics for The Times, focusing on breaking news and the 2024 campaign. More about Chris Cameron