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In statistics, hypothesis tests are used to test whether or not some hypothesis about a population parameter is true.
To perform a hypothesis test in the real world, researchers will obtain a random sample from the population and perform a hypothesis test on the sample data, using a null and alternative hypothesis:
If the p-value of the hypothesis test is less than some significance level (e.g. α = .05), then we can reject the null hypothesis and conclude that we have sufficient evidence to say that the alternative hypothesis is true.
The following examples provide several situations where hypothesis tests are used in the real world.
Hypothesis tests are often used in biology to determine whether some new treatment, fertilizer, pesticide, chemical, etc. causes increased growth, stamina, immunity, etc. in plants or animals.
For example, suppose a biologist believes that a certain fertilizer will cause plants to grow more during a one-month period than they normally do, which is currently 20 inches. To test this, she applies the fertilizer to each of the plants in her laboratory for one month.
She then performs a hypothesis test using the following hypotheses:
If the p-value of the test is less than some significance level (e.g. α = .05), then she can reject the null hypothesis and conclude that the fertilizer leads to increased plant growth.
Hypothesis tests are often used in clinical trials to determine whether some new treatment, drug, procedure, etc. causes improved outcomes in patients.
For example, suppose a doctor believes that a new drug is able to reduce blood pressure in obese patients. To test this, he may measure the blood pressure of 40 patients before and after using the new drug for one month.
He then performs a hypothesis test using the following hypotheses:
If the p-value of the test is less than some significance level (e.g. α = .05), then he can reject the null hypothesis and conclude that the new drug leads to reduced blood pressure.
Hypothesis tests are often used in business to determine whether or not some new advertising campaign, marketing technique, etc. causes increased sales.
For example, suppose a company believes that spending more money on digital advertising leads to increased sales. To test this, the company may increase money spent on digital advertising during a two-month period and collect data to see if overall sales have increased.
They may perform a hypothesis test using the following hypotheses:
If the p-value of the test is less than some significance level (e.g. α = .05), then the company can reject the null hypothesis and conclude that increased digital advertising leads to increased sales.
Hypothesis tests are also used often in manufacturing plants to determine if some new process, technique, method, etc. causes a change in the number of defective products produced.
For example, suppose a certain manufacturing plant wants to test whether or not some new method changes the number of defective widgets produced per month, which is currently 250. To test this, they may measure the mean number of defective widgets produced before and after using the new method for one month.
They can then perform a hypothesis test using the following hypotheses:
If the p-value of the test is less than some significance level (e.g. α = .05), then the plant can reject the null hypothesis and conclude that the new method leads to a change in the number of defective widgets produced per month.
Introduction to Hypothesis Testing Introduction to the One Sample t-test Introduction to the Two Sample t-test Introduction to the Paired Samples t-test
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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.
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]
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
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
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
Read: Research Report: Definition, Types + [Writing Guide]
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
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
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
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
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.
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.
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To successfully confirm or refute an assumption, the researcher goes through five (5) stages of hypothesis testing;
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.
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.
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]
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:
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.
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.
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.
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:
Several limitations of hypothesis testing can affect the quality of data you get from this process. Some of these limitations include:
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Simple guide on pure or basic research, its methods, characteristics, advantages, and examples in science, medicine, education and psychology
In this article, we will discuss the concept of internal validity, some clear examples, its importance, and how to test it.
This article will discuss the two different types of errors in hypothesis testing and how you can prevent them from occurring in your research
We are going to discuss alternative hypotheses and null hypotheses in this post and how they work in research.
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The previous two chapters introduced methods for organizing and summarizing sample data, and using sample statistics to estimate population parameters. This chapter introduces the next major topic of inferential statistics: hypothesis testing.
A hypothesis is a statement or claim about a property of a population.
When conducting scientific research, typically there is some known information, perhaps from some past work or from a long accepted idea. We want to test whether this claim is believable. This is the basic idea behind a hypothesis test:
For example, past research tells us that the average life span for a hummingbird is about four years. You have been studying the hummingbirds in the southeastern United States and find a sample mean lifespan of 4.8 years. Should you reject the known or accepted information in favor of your results? How confident are you in your estimate? At what point would you say that there is enough evidence to reject the known information and support your alternative claim? How far from the known mean of four years can the sample mean be before we reject the idea that the average lifespan of a hummingbird is four years?
Hypothesis testing is a procedure, based on sample evidence and probability, used to test claims regarding a characteristic of a population.
A hypothesis is a claim or statement about a characteristic of a population of interest to us. A hypothesis test is a way for us to use our sample statistics to test a specific claim.
The population mean weight is known to be 157 lb. We want to test the claim that the mean weight has increased.
Two years ago, the proportion of infected plants was 37%. We believe that a treatment has helped, and we want to test the claim that there has been a reduction in the proportion of infected plants.
The null hypothesis is a statement about the value of a population parameter, such as the population mean (µ) or the population proportion ( p ). It contains the condition of equality and is denoted as H 0 (H-naught).
H 0 : µ = 157 or H 0 : p = 0.37
The alternative hypothesis is the claim to be tested, the opposite of the null hypothesis. It contains the value of the parameter that we consider plausible and is denoted as H 1 .
H 1 : µ > 157 or H 1 : p ≠ 0.37
The test statistic is a value computed from the sample data that is used in making a decision about the rejection of the null hypothesis. The test statistic converts the sample mean ( x̄ ) or sample proportion ( p̂ ) to a Z- or t-score under the assumption that the null hypothesis is true . It is used to decide whether the difference between the sample statistic and the hypothesized claim is significant.
The p-value is the area under the curve to the left or right of the test statistic. It is compared to the level of significance ( α ).
The critical value is the value that defines the rejection zone (the test statistic values that would lead to rejection of the null hypothesis). It is defined by the level of significance.
The level of significance ( α ) is the probability that the test statistic will fall into the critical region when the null hypothesis is true. This level is set by the researcher.
The conclusion is the final decision of the hypothesis test. The conclusion must always be clearly stated, communicating the decision based on the components of the test. It is important to realize that we never prove or accept the null hypothesis. We are merely saying that the sample evidence is not strong enough to warrant the rejection of the null hypothesis. The conclusion is made up of two parts:
1) Reject or fail to reject the null hypothesis, and 2) there is or is not enough evidence to support the alternative claim.
Option 1) Reject the null hypothesis (H 0 ). This means that you have enough statistical evidence to support the alternative claim (H 1 ).
Option 2) Fail to reject the null hypothesis (H 0 ). This means that you do NOT have enough evidence to support the alternative claim (H 1 ).
Another way to think about hypothesis testing is to compare it to the US justice system. A defendant is innocent until proven guilty (Null hypothesis—innocent). The prosecuting attorney tries to prove that the defendant is guilty (Alternative hypothesis—guilty). There are two possible conclusions that the jury can reach. First, the defendant is guilty (Reject the null hypothesis). Second, the defendant is not guilty (Fail to reject the null hypothesis). This is NOT the same thing as saying the defendant is innocent! In the first case, the prosecutor had enough evidence to reject the null hypothesis (innocent) and support the alternative claim (guilty). In the second case, the prosecutor did NOT have enough evidence to reject the null hypothesis (innocent) and support the alternative claim of guilty.
There are three different pairs of null and alternative hypotheses:
where c is some known value.
This tests whether the population parameter is equal to, versus not equal to, some specific value.
H o : μ = 12 vs. H 1 : μ ≠ 12
The critical region is divided equally into the two tails and the critical values are ± values that define the rejection zones.
A forester studying diameter growth of red pine believes that the mean diameter growth will be different if a fertilization treatment is applied to the stand.
This is a two-sided question, as the forester doesn’t state whether population mean diameter growth will increase or decrease.
This tests whether the population parameter is equal to, versus greater than, some specific value.
H o : μ = 12 vs. H 1 : μ > 12
The critical region is in the right tail and the critical value is a positive value that defines the rejection zone.
A biologist believes that there has been an increase in the mean number of lakes infected with milfoil, an invasive species, since the last study five years ago.
This is a right-sided question, as the biologist believes that there has been an increase in population mean number of infected lakes.
This tests whether the population parameter is equal to, versus less than, some specific value.
H o : μ = 12 vs. H 1 : μ < 12
The critical region is in the left tail and the critical value is a negative value that defines the rejection zone.
A scientist’s research indicates that there has been a change in the proportion of people who support certain environmental policies. He wants to test the claim that there has been a reduction in the proportion of people who support these policies.
This is a left-sided question, as the scientist believes that there has been a reduction in the true population proportion.
When the observed results (the sample statistics) are unlikely (a low probability) under the assumption that the null hypothesis is true, we say that the result is statistically significant, and we reject the null hypothesis. This result depends on the level of significance, the sample statistic, sample size, and whether it is a one- or two-sided alternative hypothesis.
When testing, we arrive at a conclusion of rejecting the null hypothesis or failing to reject the null hypothesis. Such conclusions are sometimes correct and sometimes incorrect (even when we have followed all the correct procedures). We use incomplete sample data to reach a conclusion and there is always the possibility of reaching the wrong conclusion. There are four possible conclusions to reach from hypothesis testing. Of the four possible outcomes, two are correct and two are NOT correct.
A Type I error is when we reject the null hypothesis when it is true. The symbol α (alpha) is used to represent Type I errors. This is the same alpha we use as the level of significance. By setting alpha as low as reasonably possible, we try to control the Type I error through the level of significance.
A Type II error is when we fail to reject the null hypothesis when it is false. The symbol β (beta) is used to represent Type II errors.
In general, Type I errors are considered more serious. One step in the hypothesis test procedure involves selecting the significance level ( α ), which is the probability of rejecting the null hypothesis when it is correct. So the researcher can select the level of significance that minimizes Type I errors. However, there is a mathematical relationship between α, β , and n (sample size).
The natural inclination is to select the smallest possible value for α, thinking to minimize the possibility of causing a Type I error. Unfortunately, this forces an increase in Type II errors. By making the rejection zone too small, you may fail to reject the null hypothesis, when, in fact, it is false. Typically, we select the best sample size and level of significance, automatically setting β .
A Type II error ( β ) is the probability of failing to reject a false null hypothesis. It follows that 1- β is the probability of rejecting a false null hypothesis. This probability is identified as the power of the test, and is often used to gauge the test’s effectiveness in recognizing that a null hypothesis is false.
The probability that at a fixed level α significance test will reject H 0 , when a particular alternative value of the parameter is true is called the power of the test.
Power is also directly linked to sample size. For example, suppose the null hypothesis is that the mean fish weight is 8.7 lb. Given sample data, a level of significance of 5%, and an alternative weight of 9.2 lb., we can compute the power of the test to reject μ = 8.7 lb. If we have a small sample size, the power will be low. However, increasing the sample size will increase the power of the test. Increasing the level of significance will also increase power. A 5% test of significance will have a greater chance of rejecting the null hypothesis than a 1% test because the strength of evidence required for the rejection is less. Decreasing the standard deviation has the same effect as increasing the sample size: there is more information about μ .
We are going to examine two equivalent ways to perform a hypothesis test: the classical approach and the p-value approach. The classical approach is based on standard deviations. This method compares the test statistic (Z-score) to a critical value (Z-score) from the standard normal table. If the test statistic falls in the rejection zone, you reject the null hypothesis. The p-value approach is based on area under the normal curve. This method compares the area associated with the test statistic to alpha ( α ), the level of significance (which is also area under the normal curve). If the p-value is less than alpha, you would reject the null hypothesis.
As a past student poetically said: If the p-value is a wee value, Reject Ho
Both methods must have:
There are four steps required for a hypothesis test:
A forester studying diameter growth of red pine believes that the mean diameter growth will be different from the known mean growth of 1.35 inches/year if a fertilization treatment is applied to the stand. He conducts his experiment, collects data from a sample of 32 plots, and gets a sample mean diameter growth of 1.6 in./year. The population standard deviation for this stand is known to be 0.46 in./year. Does he have enough evidence to support his claim?
Step 1) State the null and alternative hypotheses.
Step 2) State the level of significance and the critical value.
Step 3) Compute the test statistic.
Step 4) State a conclusion.
In this problem, the test statistic falls in the red rejection zone. The test statistic of 3.07 is greater than the critical value of 1.96.We will reject the null hypothesis. We have enough evidence to support the claim that the mean diameter growth is different from (not equal to) 1.35 in./year.
A researcher believes that there has been an increase in the average farm size in his state since the last study five years ago. The previous study reported a mean size of 450 acres with a population standard deviation ( σ ) of 167 acres. He samples 45 farms and gets a sample mean of 485.8 acres. Is there enough information to support his claim?
We fail to reject the null hypothesis. We do not have enough evidence to support the claim that the mean farm size has increased from 450 acres.
A researcher believes that there has been a reduction in the mean number of hours that college students spend preparing for final exams. A national study stated that students at a 4-year college spend an average of 23 hours preparing for 5 final exams each semester with a population standard deviation of 7.3 hours. The researcher sampled 227 students and found a sample mean study time of 19.6 hours. Does this indicate that the average study time for final exams has decreased? Use a 1% level of significance to test this claim.
We reject the null hypothesis. We have sufficient evidence to support the claim that the mean final exam study time has decreased below 23 hours.
The p-value is the probability of observing our sample mean given that the null hypothesis is true. It is the area under the curve to the left or right of the test statistic. If the probability of observing such a sample mean is very small (less than the level of significance), we would reject the null hypothesis. Computations for the p-value depend on whether it is a one- or two-sided test.
Steps for a hypothesis test using p-values:
Instead of comparing Z-score test statistic to Z-score critical value, as in the classical method, we compare area of the test statistic to area of the level of significance.
The Decision Rule: If the p-value is less than alpha, we reject the null hypothesis
If it is a two-sided test (the alternative claim is ≠), the p-value is equal to two times the probability of the absolute value of the test statistic. If the test is a left-sided test (the alternative claim is “<”), then the p-value is equal to the area to the left of the test statistic. If the test is a right-sided test (the alternative claim is “>”), then the p-value is equal to the area to the right of the test statistic.
Let’s look at Example 6 again.
A forester studying diameter growth of red pine believes that the mean diameter growth will be different from the known mean growth of 1.35 in./year if a fertilization treatment is applied to the stand. He conducts his experiment, collects data from a sample of 32 plots, and gets a sample mean diameter growth of 1.6 in./year. The population standard deviation for this stand is known to be 0.46 in./year. Does he have enough evidence to support his claim?
Step 2) State the level of significance.
The p-value is two times the area of the absolute value of the test statistic (because the alternative claim is “not equal”).
Step 4) Compare the p-value to alpha and state a conclusion.
Let’s look at Example 7 again.
The p-value is the area to the right of the Z-score 1.44 (the hatched area).
We fail to reject the null hypothesis. We do not have enough evidence to support the claim that the mean farm size has increased.
Let’s look at Example 8 again.
The p-value is the area to the left of the test statistic (the little black area to the left of -7.02). The Z-score of -7.02 is not on the standard normal table. The smallest probability on the table is 0.0002. We know that the area for the Z-score -7.02 is smaller than this area (probability). Therefore, the p-value is <0.0002.
We reject the null hypothesis. We have enough evidence to support the claim that the mean final exam study time has decreased below 23 hours.
Both the classical method and p-value method for testing a hypothesis will arrive at the same conclusion. In the classical method, the critical Z-score is the number on the z-axis that defines the level of significance ( α ). The test statistic converts the sample mean to units of standard deviation (a Z-score). If the test statistic falls in the rejection zone defined by the critical value, we will reject the null hypothesis. In this approach, two Z-scores, which are numbers on the z-axis, are compared. In the p-value approach, the p-value is the area associated with the test statistic. In this method, we compare α (which is also area under the curve) to the p-value. If the p-value is less than α , we reject the null hypothesis. The p-value is the probability of observing such a sample mean when the null hypothesis is true. If the probability is too small (less than the level of significance), then we believe we have enough statistical evidence to reject the null hypothesis and support the alternative claim.
(referring to Ex. 8)
Test of mu = 23 vs. < 23 |
The assumed standard deviation = 7.3 |
99% Upper | |||||
N | Mean | SE Mean | Bound | Z | P |
227 | 19.600 | 0.485 | 20.727 | -7.02 | 0.000 |
Excel does not offer 1-sample hypothesis testing.
Frequently, the population standard deviation (σ) is not known. We can estimate the population standard deviation (σ) with the sample standard deviation (s). However, the test statistic will no longer follow the standard normal distribution. We must rely on the student’s t-distribution with n-1 degrees of freedom. Because we use the sample standard deviation (s), the test statistic will change from a Z-score to a t-score.
Steps for a hypothesis test are the same that we covered in Section 2.
Just as with the hypothesis test from the previous section, the data for this test must be from a random sample and requires either that the population from which the sample was drawn be normal or that the sample size is sufficiently large (n≥30). A t-test is robust, so small departures from normality will not adversely affect the results of the test. That being said, if the sample size is smaller than 30, it is always good to verify the assumption of normality through a normal probability plot.
We will still have the same three pairs of null and alternative hypotheses and we can still use either the classical approach or the p-value approach.
Selecting the correct critical value from the student’s t-distribution table depends on three factors: the type of test (one-sided or two-sided alternative hypothesis), the sample size, and the level of significance.
For a two-sided test (“not equal” alternative hypothesis), the critical value (t α /2 ), is determined by alpha ( α ), the level of significance, divided by two, to deal with the possibility that the result could be less than OR greater than the known value.
For a one-sided test (“a less than” or “greater than” alternative hypothesis), the critical value (t α ) , is determined by alpha ( α ), the level of significance, being all in the one side.
Find the critical value you would use to test the claim that μ ≠ 112 with a sample size of 18 and a 5% level of significance.
In this case, the critical value (t α /2 ) would be 2.110. This is a two-sided question (≠) so you would divide alpha by 2 (0.05/2 = 0.025) and go down the 0.025 column to 17 degrees of freedom.
What would the critical value be if you wanted to test that μ < 112 for the same data?
In this case, the critical value would be 1.740. This is a one-sided question (<) so alpha would be divided by 1 (0.05/1 = 0.05). You would go down the 0.05 column with 17 degrees of freedom to get the correct critical value.
In 2005, the mean pH level of rain in a county in northern New York was 5.41. A biologist believes that the rain acidity has changed. He takes a random sample of 11 rain dates in 2010 and obtains the following data. Use a 1% level of significance to test his claim.
4.70, 5.63, 5.02, 5.78, 4.99, 5.91, 5.76, 5.54, 5.25, 5.18, 5.01
The sample size is small and we don’t know anything about the distribution of the population, so we examine a normal probability plot. The distribution looks normal so we will continue with our test.
The sample mean is 5.343 with a sample standard deviation of 0.397.
We will fail to reject the null hypothesis. We do not have enough evidence to support the claim that the mean rain pH has changed.
Cadmium, a heavy metal, is toxic to animals. Mushrooms, however, are able to absorb and accumulate cadmium at high concentrations. The government has set safety limits for cadmium in dry vegetables at 0.5 ppm. Biologists believe that the mean level of cadmium in mushrooms growing near strip mines is greater than the recommended limit of 0.5 ppm, negatively impacting the animals that live in this ecosystem. A random sample of 51 mushrooms gave a sample mean of 0.59 ppm with a sample standard deviation of 0.29 ppm. Use a 5% level of significance to test the claim that the mean cadmium level is greater than the acceptable limit of 0.5 ppm.
The sample size is greater than 30 so we are assured of a normal distribution of the means.
Step 4) State a Conclusion.
The test statistic falls in the rejection zone. We will reject the null hypothesis. We have enough evidence to support the claim that the mean cadmium level is greater than the acceptable safe limit.
BUT, what happens if the significance level changes to 1%?
The critical value is now found by going down the 0.01 column with 50 degrees of freedom. The critical value is 2.403. The test statistic is now LESS THAN the critical value. The test statistic does not fall in the rejection zone. The conclusion will change. We do NOT have enough evidence to support the claim that the mean cadmium level is greater than the acceptable safe limit of 0.5 ppm.
The level of significance is the probability that you, as the researcher, set to decide if there is enough statistical evidence to support the alternative claim. It should be set before the experiment begins.
We can also use the p-value approach for a hypothesis test about the mean when the population standard deviation ( σ ) is unknown. However, when using a student’s t-table, we can only estimate the range of the p-value, not a specific value as when using the standard normal table. The student’s t-table has area (probability) across the top row in the table, with t-scores in the body of the table.
Estimating P-value from a Student’s T-table
If your test statistic is 3.789 with 3 degrees of freedom, you would go across the 3 df row. The value 3.789 falls between the values 3.482 and 4.541 in that row. Therefore, the p-value is between 0.02 and 0.01. The p-value will be greater than 0.01 but less than 0.02 (0.01<p<0.02).
If your level of significance is 5%, you would reject the null hypothesis as the p-value (0.01-0.02) is less than alpha ( α ) of 0.05.
If your level of significance is 1%, you would fail to reject the null hypothesis as the p-value (0.01-0.02) is greater than alpha ( α ) of 0.01.
Software packages typically output p-values. It is easy to use the Decision Rule to answer your research question by the p-value method.
(referring to Ex. 12)
Test of mu = 0.5 vs. > 0.5
95% Lower | ||||||
N | Mean | StDev | SE Mean | Bound | T | P |
51 | 0.5900 | 0.2900 | 0.0406 | 0.5219 | 2.22 | 0.016 |
Additional example: www.youtube.com/watch?v=WwdSjO4VUsg .
Frequently, the parameter we are testing is the population proportion.
Recall that the best point estimate of p , the population proportion, is given by
when np (1 – p )≥10. We can use both the classical approach and the p-value approach for testing.
The steps for a hypothesis test are the same that we covered in Section 2.
The test statistic follows the standard normal distribution. Notice that the standard error (the denominator) uses p instead of p̂ , which was used when constructing a confidence interval about the population proportion. In a hypothesis test, the null hypothesis is assumed to be true, so the known proportion is used.
A botanist has produced a new variety of hybrid soy plant that is better able to withstand drought than other varieties. The botanist knows the seed germination for the parent plants is 75%, but does not know the seed germination for the new hybrid. He tests the claim that it is different from the parent plants. To test this claim, 450 seeds from the hybrid plant are tested and 321 have germinated. Use a 5% level of significance to test this claim that the germination rate is different from 75%.
This is a two-sided question so alpha is divided by 2.
The test statistic does not fall in the rejection zone. We fail to reject the null hypothesis. We do not have enough evidence to support the claim that the germination rate of the hybrid plant is different from the parent plants.
Let’s answer this question using the p-value approach. Remember, for a two-sided alternative hypothesis (“not equal”), the p-value is two times the area of the test statistic. The test statistic is -1.81 and we want to find the area to the left of -1.81 from the standard normal table.
Now compare the p-value to alpha. The Decision Rule states that if the p-value is less than alpha, reject the H 0 . In this case, the p-value (0.0702) is greater than alpha (0.05) so we will fail to reject H 0 . We do not have enough evidence to support the claim that the germination rate of the hybrid plant is different from the parent plants.
You are a biologist studying the wildlife habitat in the Monongahela National Forest. Cavities in older trees provide excellent habitat for a variety of birds and small mammals. A study five years ago stated that 32% of the trees in this forest had suitable cavities for this type of wildlife. You believe that the proportion of cavity trees has increased. You sample 196 trees and find that 79 trees have cavities. Does this evidence support your claim that there has been an increase in the proportion of cavity trees?
Use a 10% level of significance to test this claim.
This is a one-sided question so alpha is divided by 1.
The test statistic is larger than the critical value (it falls in the rejection zone). We will reject the null hypothesis. We have enough evidence to support the claim that there has been an increase in the proportion of cavity trees.
Now use the p-value approach to answer the question. This is a right-sided question (“greater than”), so the p-value is equal to the area to the right of the test statistic. Go to the positive side of the standard normal table and find the area associated with the Z-score of 2.49. The area is 0.9936. Remember that this table is cumulative from the left. To find the area to the right of 2.49, we subtract from one.
p-value = (1 – 0.9936) = 0.0064
The p-value is less than the level of significance (0.10), so we reject the null hypothesis. We have enough evidence to support the claim that the proportion of cavity trees has increased.
(referring to Ex. 15)
Test of p = 0.32 vs. p > 0.32
90% Lower | ||||||
Sample | X | N | Sample p | Bound | Z-Value | p-Value |
1 | 79 | 196 | 0.403061 | 0.358160 | 2.49 | 0.006 |
Using the normal approximation. |
When people think of statistical inference, they usually think of inferences involving population means or proportions. However, the particular population parameter needed to answer an experimenter’s practical questions varies from one situation to another, and sometimes a population’s variability is more important than its mean. Thus, product quality is often defined in terms of low variability.
Sample variance S 2 can be used for inferences concerning a population variance σ 2 . For a random sample of n measurements drawn from a normal population with mean μ and variance σ 2 , the value S 2 provides a point estimate for σ 2 . In addition, the quantity ( n – 1) S 2 / σ 2 follows a Chi-square ( χ 2 ) distribution, with df = n – 1.
The properties of Chi-square ( χ 2 ) distribution are:
Alternative hypothesis:
where the χ 2 critical value in the rejection region is based on degrees of freedom df = n – 1 and a specified significance level of α .
As with previous sections, if the test statistic falls in the rejection zone set by the critical value, you will reject the null hypothesis.
A forester wants to control a dense understory of striped maple that is interfering with desirable hardwood regeneration using a mist blower to apply an herbicide treatment. She wants to make sure that treatment has a consistent application rate, in other words, low variability not exceeding 0.25 gal./acre (0.06 gal. 2 ). She collects sample data (n = 11) on this type of mist blower and gets a sample variance of 0.064 gal. 2 Using a 5% level of significance, test the claim that the variance is significantly greater than 0.06 gal. 2
H 0 : σ 2 = 0.06
H 1 : σ 2 >0.06
The critical value is 18.307. Any test statistic greater than this value will cause you to reject the null hypothesis.
The test statistic is
We fail to reject the null hypothesis. The forester does NOT have enough evidence to support the claim that the variance is greater than 0.06 gal. 2 You can also estimate the p-value using the same method as for the student t-table. Go across the row for degrees of freedom until you find the two values that your test statistic falls between. In this case going across the row 10, the two table values are 4.865 and 15.987. Now go up those two columns to the top row to estimate the p-value (0.1-0.9). The p-value is greater than 0.1 and less than 0.9. Both are greater than the level of significance (0.05) causing us to fail to reject the null hypothesis.
(referring to Ex. 16)
Test and CI for One Variance
Method | ||
Null hypothesis | Sigma-squared | = 0.06 |
Alternative hypothesis | Sigma-squared | > 0.06 |
The chi-square method is only for the normal distribution.
Test | |||
Method | Statistic | DF | P-Value |
Chi-Square | 10.67 | 10 | 0.384 |
Excel does not offer 1-sample χ 2 testing.
To test a claim about μ when σ is known.
Natural Resources Biometrics Copyright © 2014 by Diane Kiernan is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License , except where otherwise noted.
Hypothesis Definition, Format, Examples, and Tips
Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."
Amy Morin, LCSW, is a psychotherapist and international bestselling author. Her books, including "13 Things Mentally Strong People Don't Do," have been translated into more than 40 languages. Her TEDx talk, "The Secret of Becoming Mentally Strong," is one of the most viewed talks of all time.
Verywell / Alex Dos Diaz
Falsifiability of a hypothesis.
Hypotheses examples.
A hypothesis is a tentative statement about the relationship between two or more variables. It is a specific, testable prediction about what you expect to happen in a study. It is a preliminary answer to your question that helps guide the research process.
Consider a study designed to examine the relationship between sleep deprivation and test performance. The hypothesis might be: "This study is designed to assess the hypothesis that sleep-deprived people will perform worse on a test than individuals who are not sleep-deprived."
A hypothesis is crucial to scientific research because it offers a clear direction for what the researchers are looking to find. This allows them to design experiments to test their predictions and add to our scientific knowledge about the world. This article explores how a hypothesis is used in psychology research, how to write a good hypothesis, and the different types of hypotheses you might use.
In the scientific method , whether it involves research in psychology, biology, or some other area, a hypothesis represents what the researchers think will happen in an experiment. The scientific method involves the following steps:
The hypothesis is a prediction, but it involves more than a guess. Most of the time, the hypothesis begins with a question which is then explored through background research. At this point, researchers then begin to develop a testable hypothesis.
Unless you are creating an exploratory study, your hypothesis should always explain what you expect to happen.
In a study exploring the effects of a particular drug, the hypothesis might be that researchers expect the drug to have some type of effect on the symptoms of a specific illness. In psychology, the hypothesis might focus on how a certain aspect of the environment might influence a particular behavior.
Remember, a hypothesis does not have to be correct. While the hypothesis predicts what the researchers expect to see, the goal of the research is to determine whether this guess is right or wrong. When conducting an experiment, researchers might explore numerous factors to determine which ones might contribute to the ultimate outcome.
In many cases, researchers may find that the results of an experiment do not support the original hypothesis. When writing up these results, the researchers might suggest other options that should be explored in future studies.
In many cases, researchers might draw a hypothesis from a specific theory or build on previous research. For example, prior research has shown that stress can impact the immune system. So a researcher might hypothesize: "People with high-stress levels will be more likely to contract a common cold after being exposed to the virus than people who have low-stress levels."
In other instances, researchers might look at commonly held beliefs or folk wisdom. "Birds of a feather flock together" is one example of folk adage that a psychologist might try to investigate. The researcher might pose a specific hypothesis that "People tend to select romantic partners who are similar to them in interests and educational level."
So how do you write a good hypothesis? When trying to come up with a hypothesis for your research or experiments, ask yourself the following questions:
Before you come up with a specific hypothesis, spend some time doing background research. Once you have completed a literature review, start thinking about potential questions you still have. Pay attention to the discussion section in the journal articles you read . Many authors will suggest questions that still need to be explored.
To form a hypothesis, you should take these steps:
In the scientific method , falsifiability is an important part of any valid hypothesis. In order to test a claim scientifically, it must be possible that the claim could be proven false.
Students sometimes confuse the idea of falsifiability with the idea that it means that something is false, which is not the case. What falsifiability means is that if something was false, then it is possible to demonstrate that it is false.
One of the hallmarks of pseudoscience is that it makes claims that cannot be refuted or proven false.
A variable is a factor or element that can be changed and manipulated in ways that are observable and measurable. However, the researcher must also define how the variable will be manipulated and measured in the study.
Operational definitions are specific definitions for all relevant factors in a study. This process helps make vague or ambiguous concepts detailed and measurable.
For example, a researcher might operationally define the variable " test anxiety " as the results of a self-report measure of anxiety experienced during an exam. A "study habits" variable might be defined by the amount of studying that actually occurs as measured by time.
These precise descriptions are important because many things can be measured in various ways. Clearly defining these variables and how they are measured helps ensure that other researchers can replicate your results.
One of the basic principles of any type of scientific research is that the results must be replicable.
Replication means repeating an experiment in the same way to produce the same results. By clearly detailing the specifics of how the variables were measured and manipulated, other researchers can better understand the results and repeat the study if needed.
Some variables are more difficult than others to define. For example, how would you operationally define a variable such as aggression ? For obvious ethical reasons, researchers cannot create a situation in which a person behaves aggressively toward others.
To measure this variable, the researcher must devise a measurement that assesses aggressive behavior without harming others. The researcher might utilize a simulated task to measure aggressiveness in this situation.
The hypothesis you use will depend on what you are investigating and hoping to find. Some of the main types of hypotheses that you might use include:
A hypothesis often follows a basic format of "If {this happens} then {this will happen}." One way to structure your hypothesis is to describe what will happen to the dependent variable if you change the independent variable .
The basic format might be: "If {these changes are made to a certain independent variable}, then we will observe {a change in a specific dependent variable}."
Once a researcher has formed a testable hypothesis, the next step is to select a research design and start collecting data. The research method depends largely on exactly what they are studying. There are two basic types of research methods: descriptive research and experimental research.
Descriptive research such as case studies , naturalistic observations , and surveys are often used when conducting an experiment is difficult or impossible. These methods are best used to describe different aspects of a behavior or psychological phenomenon.
Once a researcher has collected data using descriptive methods, a correlational study can examine how the variables are related. This research method might be used to investigate a hypothesis that is difficult to test experimentally.
Experimental methods are used to demonstrate causal relationships between variables. In an experiment, the researcher systematically manipulates a variable of interest (known as the independent variable) and measures the effect on another variable (known as the dependent variable).
Unlike correlational studies, which can only be used to determine if there is a relationship between two variables, experimental methods can be used to determine the actual nature of the relationship—whether changes in one variable actually cause another to change.
The hypothesis is a critical part of any scientific exploration. It represents what researchers expect to find in a study or experiment. In situations where the hypothesis is unsupported by the research, the research still has value. Such research helps us better understand how different aspects of the natural world relate to one another. It also helps us develop new hypotheses that can then be tested in the future.
Thompson WH, Skau S. On the scope of scientific hypotheses . R Soc Open Sci . 2023;10(8):230607. doi:10.1098/rsos.230607
Taran S, Adhikari NKJ, Fan E. Falsifiability in medicine: what clinicians can learn from Karl Popper [published correction appears in Intensive Care Med. 2021 Jun 17;:]. Intensive Care Med . 2021;47(9):1054-1056. doi:10.1007/s00134-021-06432-z
Eyler AA. Research Methods for Public Health . 1st ed. Springer Publishing Company; 2020. doi:10.1891/9780826182067.0004
Nosek BA, Errington TM. What is replication ? PLoS Biol . 2020;18(3):e3000691. doi:10.1371/journal.pbio.3000691
Aggarwal R, Ranganathan P. Study designs: Part 2 - Descriptive studies . Perspect Clin Res . 2019;10(1):34-36. doi:10.4103/picr.PICR_154_18
Nevid J. Psychology: Concepts and Applications. Wadworth, 2013.
By Kendra Cherry, MSEd Kendra Cherry, MS, is a psychosocial rehabilitation specialist, psychology educator, and author of the "Everything Psychology Book."
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Hypothesis testing is the process of making a choice between two conflicting hypotheses. The null hypothesis, H0, is a statistical proposition stating that there is no significant difference between a hypothesized value of a population parameter and its value estimated from a sample drawn from that population. The alternative hypothesis, H1 or Ha, is a statistical proposition stating that there is a significant difference between a hypothesized value of a population parameter and its estimated value. When the null hypothesis is tested, a decision is either correct or incorrect. An incorrect decision can be made in two ways: We can reject the null hypothesis when it is true (Type I error) or we can fail to reject the null hypothesis when it is false (Type II error). The probability of making Type I and Type II errors is designated by alpha and beta, respectively. The smallest observed significance level for which the null hypothesis would be rejected is referred to as the p-value. The p-value only has meaning as a measure of confidence when the decision is to reject the null hypothesis. It has no meaning when the decision is that the null hypothesis is true.
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The bottom line.
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.
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.
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."
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.”
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.
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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A. banerjee.
Department of Community Medicine, D. Y. Patil Medical College, Pune - 411018, India
J. s. bhawalkar.
Few clinicians grasp the true concept of probability expressed in the ‘ P value.’ For most, a statistically significant P value is the end of the search for truth. In fact, the opposite is the case. The present paper attempts to put the P value in proper perspective by explaining different types of probabilities, their role in clinical decision making, medical research and hypothesis testing.
The clinician who wishes to remain abreast with the results of medical research needs to develop a statistical sense. He reads a number of journal articles; and constantly, he must ask questions such as, “Am I convinced that lack of mental activity predisposes to Alzheimer’s? Or “Do I believe that a particular drug cures more patients than the drug I use currently?”
The results of most studies are quantitative; and in earlier times, the reader made up his mind whether or not to accept the results of a particular study by merely looking at the figures. For instance, if 25 out of 30 patients were cured with a new drug compared with 15 out of the 30 on placebo, the superiority of the new drug was readily accepted.
In recent years, the presentation of medical research has undergone much transformation. Nowadays, no respectable journal will accept a paper if the results have not been subjected to statistical significance tests. The use of statistics has accelerated with the ready availability of statistical software. It has now become fashionable to organize workshops on research methodology and biostatistics. No doubt, this development was long overdue and one concedes that the methodologies of most medical papers have considerably improved in recent years. But at the same time, a new problem has arisen. The reading of medical journals today presupposes considerable statistical knowledge; however, those doctors who are not familiar with statistical theory tend to interpret the results of significance tests uncritically or even incorrectly.
It is often overlooked that the results of a statistical test depend not only on the observed data but also on the choice of statistical model. The statistician doing analysis of the data has a choice between several tests which are based on different models and assumptions. Unfortunately, many research workers who know little about statistics leave the statistical analysis to statisticians who know little about medicine; and the end result may well be a series of meaningless calculations.
Many readers of medical journals do not know the correct interpretation of ‘ P values,’ which are the results of significance tests. Usually, it is only stated whether the P value is below 5% ( P < .05) or above 5% ( P > .05). According to convention, the results of P < .05 are said to be statistically significant, and those with P > .05 are said to be statistically nonsignificant. These expressions are taken so seriously by most that it is almost considered ‘unscientific’ to believe in a nonsignificant result or not to believe in a ‘significant’ result. It is taken for granted that a ‘significant’ difference is a true difference and that a ‘nonsignificant’ difference is a chance finding and does not merit further exploration. Nothing can be further from the truth.
The present paper endeavors to explain the meaning of probability, its role in everyday clinical practice and the concepts behind hypothesis testing.
Probability is a recurring theme in medical practice. No doctor who returns home from a busy day at the hospital is spared the nagging feeling that some of his diagnoses may turn out to be wrong, or some of his treatments may not lead to the expected cure. Encountering the unexpected is an occupational hazard in clinical practice. Doctors after some experience in their profession reconcile to the fact that diagnosis and prognosis always have varying degrees of uncertainty and at best can be stated as probable in a particular case.
Critical appraisal of medical journals also leads to the same gut feeling. One is bombarded with new research results, but experience dictates that well-established facts of today may be refuted in some other scientific publication in the following weeks or months. When a practicing clinician reads that some new treatment is superior to the conventional one, he will assess the evidence critically, and at best he will conclude that probably it is true.
The statistical probability concept is so widely prevalent that almost everyone believes that probability is a frequency . It is not, of course, an ordinary frequency which can be estimated by simple observations, but it is the ideal or truth in the universe , which is reflected by the observed frequency. For example, when we want to determine the probability of obtaining an ace from a pack of cards (which, let us assume has been tampered with by a dishonest gambler), we proceed by drawing a card from the pack a large number of times, as we know in the long run, the observed frequency will approach the true probability or truth in the universe. Mathematicians often state that a probability is a long-run frequency, and a probability that is defined in this way is called a frequential probability . The exact magnitude of a frequential probability will remain elusive as we cannot make an infinite number of observations; but when we have made a decent number of observations (adequate sample size), we can calculate the confidence intervals, which are likely to include the true frequential probability. The width of the confidence interval depends on the number of observations (sample size).
The frequential probability concept is so prevalent that we tend to overlook terms like chance, risk and odds, in which the term probability implies a different meaning. Few hypothetical examples will make this clear. Consider the statement, “The cure for Alzheimer’s disease will probably be discovered in the coming decade.” This statement does not indicate the basis of this expectation or belief as in frequential probability, where a number of repeated observations provide the foundation for probability calculation. However, it may be based on the present state of research in Alzheimer’s. A probabilistic statement incorporates some amount of uncertainty, which may be quantified as follows: A politician may state that there is a fifty-fifty chance of winning the next election, a bookie may say that the odds of India winning the next one-day cricket game is four to one, and so on. At first glance, such probabilities may appear frequential ones, but a little reflection will reveal the contrary. We are concerned with unique events, i.e., the likely cure of a disease in the future, the next particular election, the next particular one-day game — and it makes no sense to apply the statistical idea that these types of probabilities are long-run frequencies. Further reflection will illustrate that these statements about probabilities of the election and one-day game are no different from the one about the cure for Alzheimer’s, apart from the fact that in the latter cases an attempt has been made to quantify the magnitude of belief in the occurrence of the event.
It follows from the above deliberations that we have 2 types of probability concepts. In the jargon of statistics, a probability is ideal or truth in the universe which lies beneath an observed frequency — such probabilities may be called frequential probabilities. In literary language, a probability is a measure of our subjective belief in the occurrence of a particular event or truth of a hypothesis. Such probabilities, which may be quantified that they look like frequential ones, are called subjective probabilities. Bayesian statistical theory also takes into account subjective probabilities (Lindley, 1973; Winkler, 1972). The following examples will try to illustrate these (rather confusing) concepts.
A young man is brought to the psychiatry OPD with history of withdrawal. He also gives history of talking to himself and giggling without cause. There is also a positive family history of schizophrenia. The consulting psychiatrist who examines the patient concludes that there is a 90% probability that this patient suffers from schizophrenia.
We ask the psychiatrist what makes him make such a statement. He may not be able to say that he knows from experience that 90% of such patients suffer from schizophrenia. The statement therefore may not be based on observed frequency. Instead, the psychiatrist states his probability based on his knowledge of the natural history of disease and the available literature regarding signs and symptoms in schizophrenia and positive family history. From this knowledge, the psychiatrist concludes that his belief in the diagnosis of schizophrenia in that particular patient is as strong as his belief in picking a black ball from a box containing 10 white and 90 black balls. The probability in this case is certainly subjective probability .
Let us consider another example: A 26-year-old married female patient who suffered from severe abdominal pain is referred to a hospital. She is also having amenorrhea for the past 4 months. The pain is located in the left lower abdomen. The gynecologist who examines her concludes that there is a 30% probability that the patient is suffering from ectopic pregnancy.
As before, we ask the gynecologist to explain on what basis the diagnosis of ectopic pregnancy is suspected. In this case the gynecologist states that he has studied a large number of successive patients with this symptom complex of lower abdominal pain with amenorrhea, and that a subsequent laparotomy revealed an ectopic pregnancy in 30% of the cases.
If we accept that the study cited is large enough to make us assume that the possibility of the observed frequency of ectopic pregnancy did not differ from the true frequential probability, it is natural to conclude that the gynecologist’s probability claim is more ‘evidence based’ than that of the psychiatrist, but again this is debatable.
In order to grasp this in proper perspective, it is necessary to note that the gynecologist stated that the probability of ectopic pregnancy in that particular patient was 30%. Therefore, we are concerned with a unique event just as the politician’s next election or India’s next one-day match. So in this case also, the probability is a subjective probability which was based on an observed frequency . One might also argue that even this probability is not good enough. We might ask the gynecologist to base his belief on a group of patients who also had the same age, height, color of hair and social background; and in the end, the reference group would be so restrictive that even the experience from a very large study would not provide the necessary information. If we went even further and required that he must base his belief on patients who in all respects resembled this particular patient, the probabilistic problem would vanish as we will be dealing with a certainty rather than a probability.
The clinician’s belief in a particular diagnosis in an individual patient may be based on the recorded experience in a group of patients, but it is still a subjective probability. It reflects not only the observed frequency of the disease in a reference group but also the clinician’s theoretical knowledge which determines the choice of reference group (Wulff, Pedersen and Rosenberg, 1986). Recorded experience is never the sole basis of clinical decision making.
The two situations described above are relatively straightforward. The physician observed a patient with a particular set of signs and symptoms and assessed the subjective probability about the diagnosis in each case. Such probabilities have been termed diagnostic probabilities (Wulff, Pedersen and Rosenberg, 1986). In practice, however, clinicians make diagnosis in a more complex manner which they themselves may be unable to analyze logically.
For instance, suppose the clinician suspects one of his patients is suffering from a rare disease named ‘D.’ He requests a suitable test to confirm the diagnosis, and suppose the test is positive for disease ‘D.’ He now wishes to assess the probability of the diagnosis being positive on the basis of this information, but perhaps the medical literature only provides the information that a positive test is seen in 70% of the patients with disease ‘D.’ However, it is also positive in 2% of patients without disease ‘D.’ How to tackle this doctor’s dilemma? First a formal analysis may be attempted, and then we can return to everyday clinical thinking. The frequential probability which the doctor found in the literature may be written in the statistical notation as follows:
P (S/D+) = .70, i.e., the probability of the presence of this particular sign (or test) given this particular disease is 70%.
P (S/D–) = .02, i.e., the probability of this particular sign given the absence of this particular disease is 2%.
However, such probabilities are of little clinical relevance. The clinical relevance is in the ‘opposite’ probability. In clinical practice, one would like to know the P (D/S), i.e., the probability of the disease in a particular patient given this positive sign. This can be estimated by means of Bayes’ Theorem (Papoulis, 1984; Lindley, 1973; Winkler, 1972). The formula of Bayes’ Theorem is reproduced below, from which it will be evident that to calculate P(D/S), we must also know the prior probability of the presence and the absence of the disease, i.e., P (D+) and P (D–).
P (D/S) = P (S/D+) P (D+) ÷ P (S/D+) P (D+) + P (S/D–) P (D–)
In the example of the disease ‘D’ above, let us assume that we estimate that prior probability of the disease being present, i.e., P (D+), is 25%; and therefore, prior probability of the absence of disease, i.e., P (D–), is 75%. Using the Bayes’ Theorem formula, we can calculate that the probability of the disease given a positive sign, i.e., P (D/S), is 92%.
We of course do not suggest that clinicians should always make calculations of this sort when confronted with a diagnostic dilemma, but they must in an intuitive way think along these lines. Clinical knowledge is to a large extent based on textbook knowledge, and ordinary textbooks do not tell the reader much about the probabilities of different diseases given different symptoms. Bayes’ Theorem guides a clinician how to use textbook knowledge for practical clinical purposes.
The practical significance of this point is illustrated by the European doctor who accepted a position at a hospital in tropical Africa. In order to prepare himself for the new job, he bought himself a large textbook of tropical medicine and studied in great detail the clinical pictures of a large number of exotic diseases. However, for several months after his arrival at the tropical hospital, his diagnostic performance was very poor, as he knew nothing about the relative frequency of all these diseases. He had to acquaint himself with the prior probability, P (D +), of the diseases in the catchment area of the hospital before he could make precise diagnoses.
The same thing happens on a smaller scale when a doctor trained at a university hospital establishes himself in general practice. At the beginning, he will suspect his patients of all sorts of rare diseases (which are common at the university hospital), but after a while he will learn to assess correctly the frequency of different diseases in the general population.
Besides predictions on individual patients, the doctor is also concerned in generalizations to the population at large or the target population. We may say that probably there may have been life at Mars. We may even quantify our belief and mention that there is 95% probability that depression responds more quickly during treatment with a particular antidepressant than during treatment with a placebo. These probabilities are again subjective probabilities rather than frequential probabilities . The last statement does not imply that 95% of depression cases respond to the particular antidepressant or that 95% of the published reports mention that the particular antidepressant is the best. It simply means that our belief in the truth of the statement is the same as our belief in picking up a red ball from a box containing 95 red balls and 5 white balls. It means that we are, however, almost not totally convinced that the average recovery time during treatment with a particular antidepressant is shorter than during placebo treatment.
The purpose of hypothesis testing is to aid the clinician in reaching a conclusion concerning the universe by examining a sample from that universe. A hypothesis may be defined as a presumption or statement about the truth in the universe. For example, a clinician may hypothesize that a certain drug may be effective in 80% of the cases of schizophrenia. It is frequently concerned about the parameters in the population about which the presumption or statement is made. It is the basis for motivating the research project. There are two types of hypotheses, research hypothesis and statistical hypothesis (Daniel, 2000; Guyatt et al ., 1995).
Hypothesis may be generated by deduction from anatomical, physiological facts or from clinical observations.
Statistical hypotheses are hypotheses that are stated in such a way that they may be evaluated by appropriate statistical techniques.
Nature of data.
The types of data that form the basis of hypothesis testing procedures must be understood, since these dictate the choice of statistical test.
These presumptions are the normality of the population distribution, equality of the standard deviations, random samples.
There are 2 statistical hypotheses involved in hypothesis testing. These should be stated a priori and explicitly. The null hypothesis is the hypothesis to be tested. It is denoted by the symbol H 0 . It is also known as the hypothesis of no difference . The null hypothesis is set up with the sole purpose of efforts to knock it down. In the testing of hypothesis, the null hypothesis is either rejected (knocked down) or not rejected (upheld). If the null hypothesis is not rejected, the interpretation is that the data is not sufficient evidence to cause rejection. If the testing process rejects the null hypothesis, the inference is that the data available to us is not compatible with the null hypothesis and by default we accept the alternative hypothesis , which in most cases is the research hypothesis. The alternative hypothesis is designated with the symbol H A .
Neither hypothesis testing nor statistical tests lead to proof. It merely indicates whether the hypothesis is supported or not supported by the available data. When we reject a null hypothesis, we do not mean it is not true but that it may be true. By default when we do not reject the null hypothesis, we should have this limitation in mind and should not convey the impression that this implies proof.
The test statistic is the statistic that is derived from the data from the sample. Evidently, many possible values of the test statistic can be computed depending on the particular sample selected. The test statistic serves as a decision maker, nothing more, nothing less, rather than proof or lack of it. The decision to reject or not to reject the null hypothesis depends on the magnitude of the test statistic.
The error committed when a true null hypothesis is rejected is called the type I error or α error . When a false null hypothesis is not rejected, we commit type II error, or β error . When we reject a null hypothesis, there is always the risk (howsoever small it may be) of committing a type I error, i.e., rejecting a true null hypothesis. On the other hand, whenever we fail to reject a null hypothesis, the risk of failing to reject a false null hypothesis, or committing a type II error, will always be present. Put in other words, the test statistic does not eliminate uncertainty (as many tend to believe); it only quantifies our uncertainty.
From the data contained in the sample, we compute a value of the test statistic and compare it with the rejection and non-rejection regions, which have to be specified in advance.
The statistical decision consists of rejecting or of not rejecting the null hypothesis. It is rejected if the computed value of the test statistic falls in the rejection region, and it is not rejected if the value falls in the non-rejection region.
If H 0 is rejected, we conclude that H A is true. If H 0 is not rejected, we conclude that H 0 may be true.
The P value is a number that tells us how unlikely our sample values are, given that the null hypothesis is true. A P value indicating that the sample results are not likely to have occurred, if the null hypothesis is true, provides reason for doubting the truth of the null hypothesis.
We must remember that, when the null hypothesis is not rejected, one should not say the null hypothesis is accepted. We should mention that the null hypothesis is “not rejected.” We avoid using the word accepted in this case because we may have committed a type II error. Since, frequently, the probability of committing error can be quite high (particularly with small sample sizes), we should not commit ourselves to accepting the null hypothesis.
With the above discussion on probability, clinical decision making and hypothesis testing in mind, let us reconsider the meaning of P values. When we come across the statement that there is statistically significant difference between two treatment regimes with P < .05, we should not interpret that there is less than 5% probability that there is no difference, and that there is 95% probability that a difference exists, as many uninformed readers tend to do. The statement that there is difference between the cure rates of two treatments is a general one, and we have already discussed that the probability of the truth of a general statement (hypothesis) is subjective , whereas the probabilities which are calculated by statisticians are frequential ones. The hypothesis that one treatment is better than the other is either true or false and cannot be interpreted in frequential terms.
To explain this further, suppose someone claims that 20 (80%) of 25 patients who received drug A were cured, compared to 12 (48%) of 25 patients who received drug B. In this case, there are two possibilities, either the null hypothesis is true, which means that the two treatments are equally effective and the observed difference arose by chance; or the null hypothesis is not true (and we accept the alternative hypothesis by default), which means that one treatment is better than the other. The clinician wants to make up his mind to what extent he believes in the truth of the alternative hypothesis (or the falsehood of the null hypothesis ). To resolve this issue, he needs the aid of statistical analysis. However, it is essential to note that the P value does not provide a direct answer. Let us assume in this case the statistician does a significance test and gets a P value = .04, meaning that the difference is statistically significant ( P < .05). But as explained earlier, this does not mean that there is a 4% probability that the null hypothesis is true and 96% chance that the alternative hypothesis is true. The P value is a frequential probability and it provides the information that there is a 4% probability of obtaining such a difference between the cure rates, if the null hypothesis is true . In other words, the statistician asks us to assume that the null hypothesis is true and to imagine that we do a large number of trials. In that case, the long-run frequency of trials which show a difference between the cure rates like the one we found, or even a larger one, will be 4%.
In order to elucidate the implications of the correct statistical definition of the P value, let us imagine that the patients who took part in the above trial suffered from depression, and that drug A was gentamycin, while drug B was a placebo. Our theoretical knowledge gives us no grounds for believing that gentamycin has any affect whatsoever in the cure of depression. For this reason, our prior confidence in the truth of the null hypothesis is immense (say, 99.99%), whereas our prior confidence in the alternative hypothesis is minute (0.01%). We must take these prior probabilities into account when we assess the result of the trial. We have the following choice. Either we accept the null hypothesis in spite of the fact that the probability of the trial result is fairly low at 4% ( P < .05) given the null hypothesis is true, or we accept the alternative hypothesis by rejecting the null hypothesis in spite of the fact that the subjective probability of that hypothesis is extremely low in the light of our prior knowledge.
It will be evident that the choice is a difficult one, as both hypotheses, each in its own way, may be said to be unlikely, but any clinician who reasons along these lines will choose that hypothesis which is least unacceptable: He will accept the null hypothesis and claim that the difference between the cure rates arose by chance (however small it may be), as he does not feel that the evidence from this single trial is sufficient to shake his prior belief in the null hypothesis.
Misinterpretation of P values is extremely common. One of the reasons may be that those who teach research methods do not themselves appreciate the problem. The P value is the probability of obtaining a value of the test statistic as large as or larger than the one computed from the data when in reality there is no difference between the different treatments. In other words, the P value is the probability of being wrong when asserting that a difference exists.
Lastly, we must remember we do not establish proof by hypothesis testing, and uncertainty will always remain in empirical research; at the most, we can only quantify our uncertainty.
Source of Support: Nil
Conflict of Interest: None declared.
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There are 5 main hypothesis testing steps, which will be outlined in this section. The steps are: Determine the null hypothesis: In this step, the statistician should identify the idea that is ...
Present the findings in your results and discussion section. Though the specific details might vary, the procedure you will use when testing a hypothesis will always follow some version of these steps. Table of contents. Step 1: State your null and alternate hypothesis. Step 2: Collect data. Step 3: Perform a statistical test.
Hypothesis Testing. Investigators conducting studies need research questions and hypotheses to guide analyses. Starting with broad research questions (RQs), investigators then identify a gap in current clinical practice or research. ... The null hypothesis is deemed true until a study presents significant data to support rejecting the null ...
Hypothesis testing involves designing a study and analyzing the data in order to see if the mean of the study significantly differs from the population mean. Designing a study can be done in three ...
A hypothesis test is a procedure used in statistics to assess whether a particular viewpoint is likely to be true. They follow a strict protocol, and they generate a 'p-value', on the basis of which a decision is made about the truth of the hypothesis under investigation.All of the routine statistical 'tests' used in research—t-tests, χ 2 tests, Mann-Whitney tests, etc.—are all ...
Hypothesis testing is a scientific method used for making a decision and drawing conclusions by using a statistical approach. It is used to suggest new ideas by testing theories to know whether or not the sample data supports research. A research hypothesis is a predictive statement that has to be tested using scientific methods that join an ...
The results of the study are then extrapolated to generate inferences about the population. We do this using a process known as hypothesis testing. This means that the results of the study may not always be identical to the results we would expect to find in the population; i.e., there is the possibility that the study results may be erroneous.
Step 7: Based on steps 5 and 6, draw a conclusion about H0. If the F\calculated F \calculated from the data is larger than the Fα F α, then you are in the rejection region and you can reject the null hypothesis with (1 − α) ( 1 − α) level of confidence. Note that modern statistical software condenses steps 6 and 7 by providing a p p -value.
Hypothesis testing is the process used to evaluate the strength of evidence from the sample and provides a framework for making determinations related to the population, ie, it provides a method for understanding how reliably one can extrapolate observed findings in a sample under study to the larger population from which the sample was drawn ...
A hypothesis test consists of five steps: 1. State the hypotheses. State the null and alternative hypotheses. These two hypotheses need to be mutually exclusive, so if one is true then the other must be false. 2. Determine a significance level to use for the hypothesis. Decide on a significance level.
Developing a hypothesis (with example) Step 1. Ask a question. Writing a hypothesis begins with a research question that you want to answer. The question should be focused, specific, and researchable within the constraints of your project. Example: Research question.
23.1 How Hypothesis Tests Are Reported in the News 1. Determine the null hypothesis and the alternative hypothesis. 2. Collect and summarize the data into a test ... Case Study 6.4: Smoking During Pregnancy and Child's IQ Study investigated impact of maternal smoking on subsequent IQ of child at ages 1, 2, 3, and 4 years of age. ...
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 ...
Example 1: Biology. Hypothesis tests are often used in biology to determine whether some new treatment, fertilizer, pesticide, chemical, etc. causes increased growth, stamina, immunity, etc. in plants or animals. For example, suppose a biologist believes that a certain fertilizer will cause plants to grow more during a one-month period than ...
In this video there was no critical value set for this experiment. In the last seconds of the video, Sal briefly mentions a p-value of 5% (0.05), which would have a critical of value of z = (+/-) 1.96. Since the experiment produced a z-score of 3, which is more extreme than 1.96, we reject the null hypothesis.
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.
A statistical hypothesis test is a method of statistical inference used to decide whether the data sufficiently support a particular hypothesis. ... Extensions to the theory of hypothesis testing include the study of the power of tests, i.e. the probability of correctly rejecting the null hypothesis given that it is false.
Formulating Hypotheses for Different Study Designs. Generating a testable working hypothesis is the first step towards conducting original research. Such research may prove or disprove the proposed hypothesis. Case reports, case series, online surveys and other observational studies, clinical trials, and narrative reviews help to generate ...
Components of a Formal Hypothesis Test. The null hypothesis is a statement about the value of a population parameter, such as the population mean (µ) or the population proportion (p).It contains the condition of equality and is denoted as H 0 (H-naught).. H 0: µ = 157 or H 0: p = 0.37. The alternative hypothesis is the claim to be tested, the opposite of the null hypothesis.
A hypothesis is a tentative statement about the relationship between two or more variables. It is a specific, testable prediction about what you expect to happen in a study. It is a preliminary answer to your question that helps guide the research process. Consider a study designed to examine the relationship between sleep deprivation and test ...
Hypothesis testing is the process of making a choice between two conflicting hypotheses. The null hypothesis, H0, is a statistical proposition stating that there is no significant difference between a hypothesized value of a population parameter and its value estimated from a sample drawn from that …
Hypothesis 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 ...
The present paper attempts to put the P value in proper perspective by explaining different types of probabilities, their role in clinical decision making, medical research and hypothesis testing. Keywords: Hypothesis testing, P value, Probability. The clinician who wishes to remain abreast with the results of medical research needs to develop ...
Statistics document from University of Fredericton, 9 pages, HYPOTHESIS TESTING NOTES 2-Tailed Test Tests if a mean or proportion is different from a theoretical value. Ho: m = 100 H1: m <> 100 Word problem might state that the mean is "equal to" 100. Word problem might state that the mean is "different from" or "n