If you are interested in further details of probability and sampling theory at this point then please refer to one of the general texts listed in the reference section .
You must understand confidence intervals if you intend to quote P values in reports and papers. Statistical referees of scientific journals expect authors to quote confidence intervals with greater prominence than P values.
Notes about Type I error :
Notes about Type II error :
Copyright © 1987-2024 Iain E. Buchan, all rights reserved. Download software here .
Hypothesis testing, the null and alternative hypothesis.
In order to undertake hypothesis testing you need to express your research hypothesis as a null and alternative hypothesis. The null hypothesis and alternative hypothesis are statements regarding the differences or effects that occur in the population. You will use your sample to test which statement (i.e., the null hypothesis or alternative hypothesis) is most likely (although technically, you test the evidence against the null hypothesis). So, with respect to our teaching example, the null and alternative hypothesis will reflect statements about all statistics students on graduate management courses.
The null hypothesis is essentially the "devil's advocate" position. That is, it assumes that whatever you are trying to prove did not happen ( hint: it usually states that something equals zero). For example, the two different teaching methods did not result in different exam performances (i.e., zero difference). Another example might be that there is no relationship between anxiety and athletic performance (i.e., the slope is zero). The alternative hypothesis states the opposite and is usually the hypothesis you are trying to prove (e.g., the two different teaching methods did result in different exam performances). Initially, you can state these hypotheses in more general terms (e.g., using terms like "effect", "relationship", etc.), as shown below for the teaching methods example:
Null Hypotheses (H ): | Undertaking seminar classes has no effect on students' performance. |
Alternative Hypothesis (H ): | Undertaking seminar class has a positive effect on students' performance. |
Depending on how you want to "summarize" the exam performances will determine how you might want to write a more specific null and alternative hypothesis. For example, you could compare the mean exam performance of each group (i.e., the "seminar" group and the "lectures-only" group). This is what we will demonstrate here, but other options include comparing the distributions , medians , amongst other things. As such, we can state:
Null Hypotheses (H ): | The mean exam mark for the "seminar" and "lecture-only" teaching methods is the same in the population. |
Alternative Hypothesis (H ): | The mean exam mark for the "seminar" and "lecture-only" teaching methods is not the same in the population. |
Now that you have identified the null and alternative hypotheses, you need to find evidence and develop a strategy for declaring your "support" for either the null or alternative hypothesis. We can do this using some statistical theory and some arbitrary cut-off points. Both these issues are dealt with next.
The level of statistical significance is often expressed as the so-called p -value . Depending on the statistical test you have chosen, you will calculate a probability (i.e., the p -value) of observing your sample results (or more extreme) given that the null hypothesis is true . Another way of phrasing this is to consider the probability that a difference in a mean score (or other statistic) could have arisen based on the assumption that there really is no difference. Let us consider this statement with respect to our example where we are interested in the difference in mean exam performance between two different teaching methods. If there really is no difference between the two teaching methods in the population (i.e., given that the null hypothesis is true), how likely would it be to see a difference in the mean exam performance between the two teaching methods as large as (or larger than) that which has been observed in your sample?
So, you might get a p -value such as 0.03 (i.e., p = .03). This means that there is a 3% chance of finding a difference as large as (or larger than) the one in your study given that the null hypothesis is true. However, you want to know whether this is "statistically significant". Typically, if there was a 5% or less chance (5 times in 100 or less) that the difference in the mean exam performance between the two teaching methods (or whatever statistic you are using) is as different as observed given the null hypothesis is true, you would reject the null hypothesis and accept the alternative hypothesis. Alternately, if the chance was greater than 5% (5 times in 100 or more), you would fail to reject the null hypothesis and would not accept the alternative hypothesis. As such, in this example where p = .03, we would reject the null hypothesis and accept the alternative hypothesis. We reject it because at a significance level of 0.03 (i.e., less than a 5% chance), the result we obtained could happen too frequently for us to be confident that it was the two teaching methods that had an effect on exam performance.
Whilst there is relatively little justification why a significance level of 0.05 is used rather than 0.01 or 0.10, for example, it is widely used in academic research. However, if you want to be particularly confident in your results, you can set a more stringent level of 0.01 (a 1% chance or less; 1 in 100 chance or less).
When considering whether we reject the null hypothesis and accept the alternative hypothesis, we need to consider the direction of the alternative hypothesis statement. For example, the alternative hypothesis that was stated earlier is:
Alternative Hypothesis (H ): | Undertaking seminar classes has a positive effect on students' performance. |
The alternative hypothesis tells us two things. First, what predictions did we make about the effect of the independent variable(s) on the dependent variable(s)? Second, what was the predicted direction of this effect? Let's use our example to highlight these two points.
Sarah predicted that her teaching method (independent variable: teaching method), whereby she not only required her students to attend lectures, but also seminars, would have a positive effect (that is, increased) students' performance (dependent variable: exam marks). If an alternative hypothesis has a direction (and this is how you want to test it), the hypothesis is one-tailed. That is, it predicts direction of the effect. If the alternative hypothesis has stated that the effect was expected to be negative, this is also a one-tailed hypothesis.
Alternatively, a two-tailed prediction means that we do not make a choice over the direction that the effect of the experiment takes. Rather, it simply implies that the effect could be negative or positive. If Sarah had made a two-tailed prediction, the alternative hypothesis might have been:
Alternative Hypothesis (H ): | Undertaking seminar classes has an effect on students' performance. |
In other words, we simply take out the word "positive", which implies the direction of our effect. In our example, making a two-tailed prediction may seem strange. After all, it would be logical to expect that "extra" tuition (going to seminar classes as well as lectures) would either have a positive effect on students' performance or no effect at all, but certainly not a negative effect. However, this is just our opinion (and hope) and certainly does not mean that we will get the effect we expect. Generally speaking, making a one-tail prediction (i.e., and testing for it this way) is frowned upon as it usually reflects the hope of a researcher rather than any certainty that it will happen. Notable exceptions to this rule are when there is only one possible way in which a change could occur. This can happen, for example, when biological activity/presence in measured. That is, a protein might be "dormant" and the stimulus you are using can only possibly "wake it up" (i.e., it cannot possibly reduce the activity of a "dormant" protein). In addition, for some statistical tests, one-tailed tests are not possible.
Let's return finally to the question of whether we reject or fail to reject the null hypothesis.
If our statistical analysis shows that the significance level is below the cut-off value we have set (e.g., either 0.05 or 0.01), we reject the null hypothesis and accept the alternative hypothesis. Alternatively, if the significance level is above the cut-off value, we fail to reject the null hypothesis and cannot accept the alternative hypothesis. You should note that you cannot accept the null hypothesis, but only find evidence against it.
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Example \(\PageIndex{7}\)
Joon believes that 50% of first-time brides in the United States are younger than their grooms. She performs a hypothesis test to determine if the percentage is the same or different from 50% . Joon samples 100 first-time brides and 53 reply that they are younger than their grooms. For the hypothesis test, she uses a 1% level of significance.
Set up the hypothesis test:
The 1% level of significance means that α = 0.01. This is a test of a single population proportion .
\(H_{0}: p = 0.50\) \(H_{a}: p \neq 0.50\)
The words "is the same or different from" tell you this is a two-tailed test.
Calculate the distribution needed:
Random variable: \(P′ =\) the percent of of first-time brides who are younger than their grooms.
Distribution for the test: The problem contains no mention of a mean. The information is given in terms of percentages. Use the distribution for P′ , the estimated proportion.
\[P' - N\left(p, \sqrt{\frac{p-q}{n}}\right)\nonumber \]
\[P' - N\left(0.5, \sqrt{\frac{0.5-0.5}{100}}\right)\nonumber \]
where \(p = 0.50, q = 1−p = 0.50\), and \(n = 100\)
Calculate the p -value using the normal distribution for proportions:
\[p\text{-value} = P(p′ < 0.47 or p′ > 0.53) = 0.5485\nonumber \]
where \[x = 53, p' = \frac{x}{n} = \frac{53}{100} = 0.53\nonumber \].
Interpretation of the \(p\text{-value})\: If the null hypothesis is true, there is 0.5485 probability (54.85%) that the sample (estimated) proportion \(p'\) is 0.53 or more OR 0.47 or less (see the graph in Figure).
\(\mu = p = 0.50\) comes from \(H_{0}\), the null hypothesis.
\(p′ = 0.53\). Since the curve is symmetrical and the test is two-tailed, the \(p′\) for the left tail is equal to \(0.50 – 0.03 = 0.47\) where \(\mu = p = 0.50\). (0.03 is the difference between 0.53 and 0.50.)
Compare \(\alpha\) and the \(p\text{-value}\):
Since \(\alpha = 0.01\) and \(p\text{-value} = 0.5485\). \(\alpha < p\text{-value}\).
Make a decision: Since \(\alpha < p\text{-value}\), you cannot reject \(H_{0}\).
Conclusion: At the 1% level of significance, the sample data do not show sufficient evidence that the percentage of first-time brides who are younger than their grooms is different from 50%.
The \(p\text{-value}\) can easily be calculated.
Press STAT and arrow over to TESTS . Press 5:1-PropZTest . Enter .5 for \(p_{0}\), 53 for \(x\) and 100 for \(n\). Arrow down to Prop and arrow to not equals \(p_{0}\). Press ENTER . Arrow down to Calculate and press ENTER . The calculator calculates the \(p\text{-value}\) (\(p = 0.5485\)) and the test statistic (\(z\)-score). Prop not equals .5 is the alternate hypothesis. Do this set of instructions again except arrow to Draw (instead of Calculate ). Press ENTER . A shaded graph appears with \(\(z\) = 0.6\) (test statistic) and \(p = 0.5485\) (\(p\text{-value}\)). Make sure when you use Draw that no other equations are highlighted in \(Y =\) and the plots are turned off.
The Type I and Type II errors are as follows:
The Type I error is to conclude that the proportion of first-time brides who are younger than their grooms is different from 50% when, in fact, the proportion is actually 50%. (Reject the null hypothesis when the null hypothesis is true).
The Type II error is there is not enough evidence to conclude that the proportion of first time brides who are younger than their grooms differs from 50% when, in fact, the proportion does differ from 50%. (Do not reject the null hypothesis when the null hypothesis is false.)
Exercise \(\PageIndex{7}\)
A teacher believes that 85% of students in the class will want to go on a field trip to the local zoo. She performs a hypothesis test to determine if the percentage is the same or different from 85%. The teacher samples 50 students and 39 reply that they would want to go to the zoo. For the hypothesis test, use a 1% level of significance.
First, determine what type of test this is, set up the hypothesis test, find the \(p\text{-value}\), sketch the graph, and state your conclusion.
Since the problem is about percentages, this is a test of single population proportions.
Because \(p > \alpha\), we fail to reject the null hypothesis. There is not sufficient evidence to suggest that the proportion of students that want to go to the zoo is not 85%.
Example \(\PageIndex{8}\)
Suppose a consumer group suspects that the proportion of households that have three cell phones is 30%. A cell phone company has reason to believe that the proportion is not 30%. Before they start a big advertising campaign, they conduct a hypothesis test. Their marketing people survey 150 households with the result that 43 of the households have three cell phones.
Set up the Hypothesis Test:
\(H_{0}: p = 0.30, H_{a}: p \neq 0.30\)
Determine the distribution needed:
The random variable is \(P′ =\) proportion of households that have three cell phones.
The distribution for the hypothesis test is \(P' - N\left(0.30, \sqrt{\frac{(0.30 \cdot 0.70)}{150}}\right)\)
Exercise 9.6.8.2
a. The value that helps determine the \(p\text{-value}\) is \(p′\). Calculate \(p′\).
a. \(p' = \frac{x}{n}\) where \(x\) is the number of successes and \(n\) is the total number in the sample.
\(x = 43, n = 150\)
\(p′ = 43150\)
Exercise 9.6.8.3
b. What is a success for this problem?
b. A success is having three cell phones in a household.
Exercise 9.6.8.4
c. What is the level of significance?
c. The level of significance is the preset \(\alpha\). Since \(\alpha\) is not given, assume that \(\alpha = 0.05\).
Exercise 9.6.8.5
d. Draw the graph for this problem. Draw the horizontal axis. Label and shade appropriately.
Calculate the \(p\text{-value}\).
d. \(p\text{-value} = 0.7216\)
Exercise 9.6.8.6
e. Make a decision. _____________(Reject/Do not reject) \(H_{0}\) because____________.
e. Assuming that \(\alpha = 0.05, \alpha < p\text{-value}\). The decision is do not reject \(H_{0}\) because there is not sufficient evidence to conclude that the proportion of households that have three cell phones is not 30%.
Exercise \(\PageIndex{8}\)
Marketers believe that 92% of adults in the United States own a cell phone. A cell phone manufacturer believes that number is actually lower. 200 American adults are surveyed, of which, 174 report having cell phones. Use a 5% level of significance. State the null and alternative hypothesis, find the p -value, state your conclusion, and identify the Type I and Type II errors.
Because \(p < 0.05\), we reject the null hypothesis. There is sufficient evidence to conclude that fewer than 92% of American adults own cell phones.
The next example is a poem written by a statistics student named Nicole Hart. The solution to the problem follows the poem. Notice that the hypothesis test is for a single population proportion. This means that the null and alternate hypotheses use the parameter \(p\). The distribution for the test is normal. The estimated proportion \(p′\) is the proportion of fleas killed to the total fleas found on Fido. This is sample information. The problem gives a preconceived \(\alpha = 0.01\), for comparison, and a 95% confidence interval computation. The poem is clever and humorous, so please enjoy it!
Example \(\PageIndex{9}\)
My dog has so many fleas,
They do not come off with ease. As for shampoo, I have tried many types Even one called Bubble Hype, Which only killed 25% of the fleas, Unfortunately I was not pleased.
I've used all kinds of soap, Until I had given up hope Until one day I saw An ad that put me in awe.
A shampoo used for dogs Called GOOD ENOUGH to Clean a Hog Guaranteed to kill more fleas.
I gave Fido a bath And after doing the math His number of fleas Started dropping by 3's! Before his shampoo I counted 42.
At the end of his bath, I redid the math And the new shampoo had killed 17 fleas. So now I was pleased.
Now it is time for you to have some fun With the level of significance being .01, You must help me figure out
Use the new shampoo or go without?
\(H_{0}: p \leq 0.25\) \(H_{a}: p > 0.25\)
In words, CLEARLY state what your random variable \(\bar{X}\) or \(P′\) represents.
\(P′ =\) The proportion of fleas that are killed by the new shampoo
State the distribution to use for the test.
\[N\left(0.25, \sqrt{\frac{(0.25){1-0.25}}{42}}\right)\nonumber \]
Test Statistic: \(z = 2.3163\)
Calculate the \(p\text{-value}\) using the normal distribution for proportions:
\[p\text{-value} = 0.0103\nonumber \]
In one to two complete sentences, explain what the p -value means for this problem.
If the null hypothesis is true (the proportion is 0.25), then there is a 0.0103 probability that the sample (estimated) proportion is 0.4048 \(\left(\frac{17}{42}\right)\) or more.
Use the previous information to sketch a picture of this situation. CLEARLY, label and scale the horizontal axis and shade the region(s) corresponding to the \(p\text{-value}\).
Indicate the correct decision (“reject” or “do not reject” the null hypothesis), the reason for it, and write an appropriate conclusion, using complete sentences.
0.01 | Do not reject \(H_{0}\) | \(\alpha < p\text{-value}\) |
Conclusion: At the 1% level of significance, the sample data do not show sufficient evidence that the percentage of fleas that are killed by the new shampoo is more than 25%.
Construct a 95% confidence interval for the true mean or proportion. Include a sketch of the graph of the situation. Label the point estimate and the lower and upper bounds of the confidence interval.
Confidence Interval: (0.26,0.55) We are 95% confident that the true population proportion p of fleas that are killed by the new shampoo is between 26% and 55%.
This test result is not very definitive since the \(p\text{-value}\) is very close to alpha. In reality, one would probably do more tests by giving the dog another bath after the fleas have had a chance to return.
Example \(\PageIndex{11}\)
In a study of 420,019 cell phone users, 172 of the subjects developed brain cancer. Test the claim that cell phone users developed brain cancer at a greater rate than that for non-cell phone users (the rate of brain cancer for non-cell phone users is 0.0340%). Since this is a critical issue, use a 0.005 significance level. Explain why the significance level should be so low in terms of a Type I error.
We will follow the four-step process.
If we commit a Type I error, we are essentially accepting a false claim. Since the claim describes cancer-causing environments, we want to minimize the chances of incorrectly identifying causes of cancer.
Figure 9.6.11.
Figure 9.6.12.
Example \(\PageIndex{12}\)
According to the US Census there are approximately 268,608,618 residents aged 12 and older. Statistics from the Rape, Abuse, and Incest National Network indicate that, on average, 207,754 rapes occur each year (male and female) for persons aged 12 and older. This translates into a percentage of sexual assaults of 0.078%. In Daviess County, KY, there were reported 11 rapes for a population of 37,937. Conduct an appropriate hypothesis test to determine if there is a statistically significant difference between the local sexual assault percentage and the national sexual assault percentage. Use a significance level of 0.01.
We will follow the four-step plan.
Figure 9.6.13.
Figure 9.6.14.
The hypothesis test itself has an established process. This can be summarized as follows:
Notice that in performing the hypothesis test, you use \(\alpha\) and not \(\beta\). \(\beta\) is needed to help determine the sample size of the data that is used in calculating the \(p\text{-value}\). Remember that the quantity \(1 – \beta\) is called the Power of the Test . A high power is desirable. If the power is too low, statisticians typically increase the sample size while keeping α the same.If the power is low, the null hypothesis might not be rejected when it should be.
Barbara Illowsky and Susan Dean (De Anza College) with many other contributing authors. Content produced by OpenStax College is licensed under a Creative Commons Attribution License 4.0 license. Download for free at http://cnx.org/contents/[email protected] .
In hypothesis testing , the level of significance is a measure of how confident you can be about rejecting the null hypothesis. This blog post will explore what hypothesis testing is and why understanding significance levels are important for your data science projects. In addition, you will also get to test your knowledge of level of significance towards the end of the blog with the help of quiz . These questions can help you test your understanding and prepare for data science / statistics interviews . Before we look into what level of significance is, let’s quickly understand what is hypothesis testing.
Table of Contents
Hypothesis testing can be defined as tests performed to evaluate whether a claim or theory about something is true or otherwise. In order to perform hypothesis tests, the following steps need to be taken:
A detailed explanation is provided in one of my related posts titled hypothesis testing explained with examples .
The level of significance is defined as the criteria or threshold value based on which one can reject the null hypothesis or fail to reject the null hypothesis. The level of significance determines whether the outcome of hypothesis testing is statistically significant or otherwise. The significance level is also called as alpha level.
Another way of looking at the level of significance is the value which represents the likelihood of making a type I error . You may recall that Type I error occurs while evaluating hypothesis testing outcomes. If you reject the null hypothesis by mistake, you end up making a Type I error. This scenario is also termed as “false positive”. Take an example of a person alleged with committing a crime. The null hypothesis is that the person is not guilty. Type I error happens when you reject the null hypothesis by mistake. Given the example, a Type I error happens when you reject the null hypothesis that the person is not guilty by mistake. The innocent person is convicted.
The level of significance can take values such as 0.1 , 0.05 , 0.01 . The most common value of the level of significance is 0.05 . The lower the value of significance level, the lesser is the chance of type I error. That would essentially mean that the experiment or hypothesis testing outcome would really need to be highly precise for one to reject the null hypothesis. The likelihood of making a type I error would be very low. However, that does increase the chances of making type II errors as you may make mistakes in failing to reject the null hypothesis. You may want to read more details in relation to type I errors and type II errors in this post – Type I errors and Type II errors in hypothesis testing
The outcome of the hypothesis testing is evaluated with the help of a p-value. If the p-value is less than the level of significance, then the hypothesis testing outcome is statistically significant. On the other hand, if the hypothesis testing outcome is not statistically significant or the p-value is more than the level of significance, then we fail to reject the null hypothesis. The same is represented in the picture below for a right-tailed test. I will be posting details on different types of tail test in future posts.
The picture below represents the concept for two-tailed hypothesis test:
For example: Let’s say that a school principal wants to find out whether extra coaching of 2 hours after school help students do better in their exams. The hypothesis would be as follows:
Now, let’s say that we conduct this experiment with 100 students and measure their scores in exams. The test statistics is computed to be z=-0.50 (p-value=0.62). Since the p-value is more than 0.05, we fail to reject the null hypothesis. There is not enough evidence to show that there’s a difference in the performance of students based on whether they get extra coaching.
While performing hypothesis tests or experiments, it is important to keep the level of significance in mind.
In hypothesis tests, if we do not have some sort of threshold by which to determine whether your results are statistically significant enough for you to reject the null hypothesis, then it would be tough for us to determine whether your findings are significant or not. This is why we take into account levels of significance when performing hypothesis tests and experiments.
Since hypothesis testing helps us in making decisions about our data, having a level of significance set up allows one to know what sort of chances their findings might have of actually being due to the null hypothesis. If you set your level of significance at 0.05 for example, it would mean that there’s only a five percent chance that the difference between groups (assuming two groups are tested) is due to random sampling error. So if we found a difference in the performance of students based on whether they take extra coaching, we would need to consider other factors that could have contributed to the difference.
This is why hypothesis testing and level of significance go hand in hand with one another: hypothesis tests help us know whether our data falls within a certain range where it’s statistically significant or not so statistically significant whereas the level of significance tells us how likely is it that our hypothesis testing results are not due to random sampling error.
The level of significance along with the test statistic and p-value formed a key part of hypothesis testing. The value that you derive from hypothesis testing depends on whether or not you accept/reject the null hypothesis, given your findings at each step. Before going into rejection vs non-rejection, let’s understand the terms better.
If the test statistic falls within the critical region, you reject the null hypothesis. This means that your findings are statistically significant and support the alternate hypothesis. The value of the p-value determines how likely it is for finding this outcome if, in fact, the null hypothesis were true. If the p-value is less than or equal to the level of significance, you reject the null hypothesis. This means that your hypothesis testing outcome was statistically significant at a certain degree and in favor of the alternate hypothesis.
If on the other hand, the p-value is greater than alpha level or significance level, then you fail to reject the null hypothesis. These findings are not statistically significant enough for one to reject the null hypothesis. The same is represented in the diagram below:
Here are some practice questions which can help you in testing your questions, and, also prepare for interviews.
#2. which of the following is looks to be inappropriate level of significance, #3. which one of the following is considered most popular choice of significance level, #4. which of the following will result in greater type ii error, #5. the p-value less than the level of significance would mean which of the following, #6. the p-value of 0.03 is statistically significant for significance level as 0.01, #7. level of significance is also called as ________, #8. the statistically significant outcome of hypothesis testing would mean which of the following, recent posts.
Oops! Check your answers again. The minimum pass percentage is 70%.
Hypothesis testing is an important statistical concept that helps us determine whether the claim made about anything is true or otherwise. The hypothesis test statistic, level of significance, and p-value all work together to help you make decisions about your data. If our hypothesis tests show enough evidence to reject the null hypothesis, then we know statistically significant findings are at hand. This post gave you ideas for how you can use hypothesis testing in your experiments by understanding what it means when someone rejects or fails to reject the null hypothesis.
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I found it very helpful. However the differences are not too understandable for me
Very Nice Explaination. Thankyiu very much,
in your case E respresent Member or Oraganization which include on e or more peers?
Such a informative post. Keep it up
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Arcu felis bibendum ut tristique et egestas quis:
Hypothesis testing.
Key Topics:
sampled from a with unknown mean μ and known variance σ . : μ = μ H : μ ≤ μ H : μ ≥ μ | : μ ≠ μ H : μ > μ H : μ < μ |
It is either likely or unlikely that we would collect the evidence we did given the initial assumption. (Note: “likely” or “unlikely” is measured by calculating a probability!)
If it is likely , then we “ do not reject ” our initial assumption. There is not enough evidence to do otherwise.
If it is unlikely , then:
In statistics, if it is unlikely, we decide to “ reject ” our initial assumption.
First, state 2 hypotheses, the null hypothesis (“H 0 ”) and the alternative hypothesis (“H A ”)
Usually the H 0 is a statement of “no effect”, or “no change”, or “chance only” about a population parameter.
While the H A , depending on the situation, is that there is a difference, trend, effect, or a relationship with respect to a population parameter.
Then, collect evidence, such as finger prints, blood spots, hair samples, carpet fibers, shoe prints, ransom notes, handwriting samples, etc. (In statistics, the data are the evidence.)
Next, you make your initial assumption.
In statistics, we always assume the null hypothesis is true .
Then, make a decision based on the available evidence.
If the observed outcome, e.g., a sample statistic, is surprising under the assumption that the null hypothesis is true, but more probable if the alternative is true, then this outcome is evidence against H 0 and in favor of H A .
An observed effect so large that it would rarely occur by chance is called statistically significant (i.e., not likely to happen by chance).
The p -value represents how likely we would be to observe such an extreme sample if the null hypothesis were true. The p -value is a probability computed assuming the null hypothesis is true, that the test statistic would take a value as extreme or more extreme than that actually observed. Since it's a probability, it is a number between 0 and 1. The closer the number is to 0 means the event is “unlikely.” So if p -value is “small,” (typically, less than 0.05), we can then reject the null hypothesis.
Significance level, α, is a decisive value for p -value. In this context, significant does not mean “important”, but it means “not likely to happened just by chance”.
α is the maximum probability of rejecting the null hypothesis when the null hypothesis is true. If α = 1 we always reject the null, if α = 0 we never reject the null hypothesis. In articles, journals, etc… you may read: “The results were significant ( p <0.05).” So if p =0.03, it's significant at the level of α = 0.05 but not at the level of α = 0.01. If we reject the H 0 at the level of α = 0.05 (which corresponds to 95% CI), we are saying that if H 0 is true, the observed phenomenon would happen no more than 5% of the time (that is 1 in 20). If we choose to compare the p -value to α = 0.01, we are insisting on a stronger evidence!
Neither decision of rejecting or not rejecting the H entails proving the null hypothesis or the alternative hypothesis. We merely state there is enough evidence to behave one way or the other. This is also always true in statistics! |
So, what kind of error could we make? No matter what decision we make, there is always a chance we made an error.
Errors in Criminal Trial:
Errors in Hypothesis Testing
Type I error (False positive): The null hypothesis is rejected when it is true.
Type II error (False negative): The null hypothesis is not rejected when it is false.
There is always a chance of making one of these errors. But, a good scientific study will minimize the chance of doing so!
The power of a statistical test is its probability of rejecting the null hypothesis if the null hypothesis is false. That is, power is the ability to correctly reject H 0 and detect a significant effect. In other words, power is one minus the type II error risk.
\(\text{Power }=1-\beta = P\left(\text{reject} H_0 | H_0 \text{is false } \right)\)
Which error is worse?
Type I = you are innocent, yet accused of cheating on the test. Type II = you cheated on the test, but you are found innocent.
This depends on the context of the problem too. But in most cases scientists are trying to be “conservative”; it's worse to make a spurious discovery than to fail to make a good one. Our goal it to increase the power of the test that is to minimize the length of the CI.
We need to keep in mind:
(see the handout). To study the tradeoffs between the sample size, α, and Type II error we can use power and operating characteristic curves.
Assume data are independently sampled from a normal distribution with unknown mean μ and known variance σ = 9. Make an initial assumption that μ = 65. Specify the hypothesis: H : μ = 65 H : μ ≠ 65 z-statistic: 3.58 z-statistic follow N(0,1) distribution
The -value, < 0.0001, indicates that, if the average height in the population is 65 inches, it is unlikely that a sample of 54 students would have an average height of 66.4630. Alpha = 0.05. Decision: -value < alpha, thus Conclude that the average height is not equal to 65. |
What type of error might we have made?
Type I error is claiming that average student height is not 65 inches, when it really is. Type II error is failing to claim that the average student height is not 65in when it is.
We rejected the null hypothesis, i.e., claimed that the height is not 65, thus making potentially a Type I error. But sometimes the p -value is too low because of the large sample size, and we may have statistical significance but not really practical significance! That's why most statisticians are much more comfortable with using CI than tests.
Based on the CI only, how do you know that you should reject the null hypothesis? The 95% CI is (65.6628,67.2631) ... What about practical and statistical significance now? Is there another reason to suspect this test, and the -value calculations? |
There is a need for a further generalization. What if we can't assume that σ is known? In this case we would use s (the sample standard deviation) to estimate σ.
If the sample is very large, we can treat σ as known by assuming that σ = s . According to the law of large numbers, this is not too bad a thing to do. But if the sample is small, the fact that we have to estimate both the standard deviation and the mean adds extra uncertainty to our inference. In practice this means that we need a larger multiplier for the standard error.
We need one-sample t -test.
: μ = μ H : μ ≤ μ H : μ ≥ μ | : μ ≠ μ H : μ > μ H : μ < μ |
Let's go back to our CNN poll. Assume we have a SRS of 1,017 adults.
We are interested in testing the following hypothesis: H 0 : p = 0.50 vs. p > 0.50
What is the test statistic?
If alpha = 0.05, what do we conclude?
We will see more details in the next lesson on proportions, then distributions, and possible tests.
Statistics By Jim
Making statistics intuitive
By Jim Frost 8 Comments
In hypothesis testing, a Type I error is a false positive while a Type II error is a false negative. In this blog post, you will learn about these two types of errors, their causes, and how to manage them.
Hypothesis tests use sample data to make inferences about the properties of a population . You gain tremendous benefits by working with random samples because it is usually impossible to measure the entire population.
However, there are tradeoffs when you use samples. The samples we use are typically a minuscule percentage of the entire population. Consequently, they occasionally misrepresent the population severely enough to cause hypothesis tests to make Type I and Type II errors.
Hypothesis testing is a procedure in inferential statistics that assesses two mutually exclusive theories about the properties of a population. For a generic hypothesis test, the two hypotheses are as follows:
The sample data must provide sufficient evidence to reject the null hypothesis and conclude that the effect exists in the population. Ideally, a hypothesis test fails to reject the null hypothesis when the effect is not present in the population, and it rejects the null hypothesis when the effect exists.
Statisticians define two types of errors in hypothesis testing. Creatively, they call these errors Type I and Type II errors. Both types of error relate to incorrect conclusions about the null hypothesis.
The table summarizes the four possible outcomes for a hypothesis test.
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Related post : How Hypothesis Tests Work: P-values and the Significance Level
Using hypothesis tests correctly improves your chances of drawing trustworthy conclusions. However, errors are bound to occur.
Unlike the fire alarm analogy, there is no sure way to determine whether an error occurred after you perform a hypothesis test. Typically, a clearer picture develops over time as other researchers conduct similar studies and an overall pattern of results appears. Seeing how your results fit in with similar studies is a crucial step in assessing your study’s findings.
Now, let’s take a look at each type of error in more depth.
When you see a p-value that is less than your significance level , you get excited because your results are statistically significant. However, it could be a type I error . The supposed effect might not exist in the population. Again, there is usually no warning when this occurs.
Why do these errors occur? It comes down to sample error. Your random sample has overestimated the effect by chance. It was the luck of the draw. This type of error doesn’t indicate that the researchers did anything wrong. The experimental design, data collection, data validity , and statistical analysis can all be correct, and yet this type of error still occurs.
Even though we don’t know for sure which studies have false positive results, we do know their rate of occurrence. The rate of occurrence for Type I errors equals the significance level of the hypothesis test, which is also known as alpha (α).
The significance level is an evidentiary standard that you set to determine whether your sample data are strong enough to reject the null hypothesis. Hypothesis tests define that standard using the probability of rejecting a null hypothesis that is actually true. You set this value based on your willingness to risk a false positive.
Related post : How to Interpret P-values Correctly
When the significance level is 0.05 and the null hypothesis is true, there is a 5% chance that the test will reject the null hypothesis incorrectly. If you set alpha to 0.01, there is a 1% of a false positive. If 5% is good, then 1% seems even better, right? As you’ll see, there is a tradeoff between Type I and Type II errors. If you hold everything else constant, as you reduce the chance for a false positive, you increase the opportunity for a false negative.
Type I errors are relatively straightforward. The math is beyond the scope of this article, but statisticians designed hypothesis tests to incorporate everything that affects this error rate so that you can specify it for your studies. As long as your experimental design is sound, you collect valid data, and the data satisfy the assumptions of the hypothesis test, the Type I error rate equals the significance level that you specify. However, if there is a problem in one of those areas, it can affect the false positive rate.
When the null hypothesis is correct for the population, the probability that a test produces a false positive equals the significance level. However, when you look at a statistically significant test result, you cannot state that there is a 5% chance that it represents a false positive.
Why is that the case? Imagine that we perform 100 studies on a population where the null hypothesis is true. If we use a significance level of 0.05, we’d expect that five of the studies will produce statistically significant results—false positives. Afterward, when we go to look at those significant studies, what is the probability that each one is a false positive? Not 5 percent but 100%!
That scenario also illustrates a point that I made earlier. The true picture becomes more evident after repeated experimentation. Given the pattern of results that are predominantly not significant, it is unlikely that an effect exists in the population.
When you perform a hypothesis test and your p-value is greater than your significance level, your results are not statistically significant. That’s disappointing because your sample provides insufficient evidence for concluding that the effect you’re studying exists in the population. However, there is a chance that the effect is present in the population even though the test results don’t support it. If that’s the case, you’ve just experienced a Type II error . The probability of making a Type II error is known as beta (β).
What causes Type II errors? Whereas Type I errors are caused by one thing, sample error, there are a host of possible reasons for Type II errors—small effect sizes, small sample sizes, and high data variability. Furthermore, unlike Type I errors, you can’t set the Type II error rate for your analysis. Instead, the best that you can do is estimate it before you begin your study by approximating properties of the alternative hypothesis that you’re studying. When you do this type of estimation, it’s called power analysis.
To estimate the Type II error rate, you create a hypothetical probability distribution that represents the properties of a true alternative hypothesis. However, when you’re performing a hypothesis test, you typically don’t know which hypothesis is true, much less the specific properties of the distribution for the alternative hypothesis. Consequently, the true Type II error rate is usually unknown!
The Type II error rate (beta) is the probability of a false negative. Therefore, the inverse of Type II errors is the probability of correctly detecting an effect. Statisticians refer to this concept as the power of a hypothesis test. Consequently, 1 – β = the statistical power. Analysts typically estimate power rather than beta directly.
If you read my post about power and sample size analysis , you know that the three factors that affect power are sample size, variability in the population, and the effect size. As you design your experiment, you can enter estimates of these three factors into statistical software and it calculates the estimated power for your test.
Suppose you perform a power analysis for an upcoming study and calculate an estimated power of 90%. For this study, the estimated Type II error rate is 10% (1 – 0.9). Keep in mind that variability and effect size are based on estimates and guesses. Consequently, power and the Type II error rate are just estimates rather than something you set directly. These estimates are only as good as the inputs into your power analysis.
Low variability and larger effect sizes decrease the Type II error rate, which increases the statistical power. However, researchers usually have less control over those aspects of a hypothesis test. Typically, researchers have the most control over sample size, making it the critical way to manage your Type II error rate. Holding everything else constant, increasing the sample size reduces the Type II error rate and increases power.
Learn more about Power in Statistics .
The graph below illustrates the two types of errors using two sampling distributions. The critical region line represents the point at which you reject or fail to reject the null hypothesis. Of course, when you perform the hypothesis test, you don’t know which hypothesis is correct. And, the properties of the distribution for the alternative hypothesis are usually unknown. However, use this graph to understand the general nature of these errors and how they are related.
The distribution on the left represents the null hypothesis. If the null hypothesis is true, you only need to worry about Type I errors, which is the shaded portion of the null hypothesis distribution. The rest of the null distribution represents the correct decision of failing to reject the null.
On the other hand, if the alternative hypothesis is true, you need to worry about Type II errors. The shaded region on the alternative hypothesis distribution represents the Type II error rate. The rest of the alternative distribution represents the probability of correctly detecting an effect—power.
Moving the critical value line is equivalent to changing the significance level. If you move the line to the left, you’re increasing the significance level (e.g., α 0.05 to 0.10). Holding everything else constant, this adjustment increases the Type I error rate while reducing the Type II error rate. Moving the line to the right reduces the significance level (e.g., α 0.05 to 0.01), which decreases the Type I error rate but increases the type II error rate.
As you’ve seen, the nature of the two types of error, their causes, and the certainty of their rates of occurrence are all very different.
A common question is whether one type of error is worse than the other? Statisticians designed hypothesis tests to control Type I errors while Type II errors are much less defined. Consequently, many statisticians state that it is better to fail to detect an effect when it exists than it is to conclude an effect exists when it doesn’t. That is to say, there is a tendency to assume that Type I errors are worse.
However, reality is more complex than that. You should carefully consider the consequences of each type of error for your specific test.
Suppose you are assessing the strength of a new jet engine part that is under consideration. Peoples lives are riding on the part’s strength. A false negative in this scenario merely means that the part is strong enough but the test fails to detect it. This situation does not put anyone’s life at risk. On the other hand, Type I errors are worse in this situation because they indicate the part is strong enough when it is not.
Now suppose that the jet engine part is already in use but there are concerns about it failing. In this case, you want the test to be more sensitive to detecting problems even at the risk of false positives. Type II errors are worse in this scenario because the test fails to recognize the problem and leaves these problematic parts in use for longer.
Using hypothesis tests effectively requires that you understand their error rates. By setting the significance level and estimating your test’s power, you can manage both error rates so they meet your requirements.
The error rates in this post are all for individual tests. If you need to perform multiple comparisons, such as comparing group means in ANOVA, you’ll need to use post hoc tests to control the experiment-wise error rate or use the Bonferroni correction .
June 4, 2024 at 2:04 pm
Very informative.
June 9, 2023 at 9:54 am
Hi Jim- I just signed up for your newsletter and this is my first question to you. I am not a statistician but work with them in my professional life as a QC consultant in biopharmaceutical development. I have a question about Type I and Type II errors in the realm of equivalence testing using two one sided difference testing (TOST). In a recent 2020 publication that I co-authored with a statistician, we stated that the probability of concluding non-equivalence when that is the truth, (which is the opposite of power, the probability of concluding equivalence when it is correct) is 1-2*alpha. This made sense to me because one uses a 90% confidence interval on a mean to evaluate whether the result is within established equivalence bounds with an alpha set to 0.05. However, it appears that specificity (1-alpha) is always the case as is power always being 1-beta. For equivalence testing the latter is 1-2*beta/2 but for specificity it stays as 1-alpha because only one of the null hypotheses in a two-sided test can fail at one time. I still see 1-2*alpha as making more sense as we show in Figure 3 of our paper which shows the white space under the distribution of the alternative hypothesis as 1-2 alpha. The paper can be downloaded as open access here if that would make my question more clear. https://bioprocessingjournal.com/index.php/article-downloads/890-vol-19-open-access-2020-defining-therapeutic-window-for-viral-vectors-a-statistical-framework-to-improve-consistency-in-assigning-product-dose-values I have consulted with other statistical colleagues and cannot get consensus so I would love your opinion and explanation! Thanks in advance!
June 10, 2023 at 1:00 am
Let me preface my response by saying that I’m not an expert in equivalence testing. But here’s my best guess about your question.
The alpha is for each of the hypothesis tests. Each one has a type I error rate of 0.05. Or, as you say, a specificity of 1-alpha. However, there are two tests so we need to consider the family-wise error rate. The formula is the following:
FWER = 1 – (1 – α)^N
Where N is the number of hypothesis tests.
For two tests, there’s a family-wise error rate of 0.0975. Or a family-wise specificity of 0.9025.
However, I believe they use 90% CI for a different reason (although it’s a very close match to the family-wise error rate). The 90% CI provides consistent results with the two one-side 95% tests. In other words, if the 90% CI is within the equivalency bounds, then the two tests will be significant. If the CI extends above the upper bound, the corresponding test won’t be significant. Etc.
However, using either rational, I’d say the overall type I error rate is about 0.1.
I hope that answers your question. And, again, I’m not an expert in this particular test.
July 18, 2022 at 5:15 am
Thank you for your valuable content. I have a question regarding correcting for multiple tests. My question is: for exactly how many tests should I correct in the scenario below?
Background: I’m testing for differences between groups A (patient group) and B (control group) in variable X. Variable X is a biological variable present in the body’s left and right side. Variable Y is a questionnaire for group A.
Step 1. Is there a significant difference within groups in the weight of left and right variable X? (I will conduct two paired sample t-tests)
If I find a significant difference in step 1, then I will conduct steps 2A and 2B. However, if I don’t find a significant difference in step 1, then I will only conduct step 2C.
Step 2A. Is there a significant difference between groups in left variable X? (I will conduct one independent sample t-test) Step 2B. Is there a significant difference between groups in right variable X? (I will conduct one independent sample t-test)
Step 2C. Is there a significant difference between groups in total variable X (left + right variable X)? (I will conduct one independent sample t-test)
If I find a significant difference in step 1, then I will conduct with steps 3A and 3B. However, if I don’t find a significant difference in step 1, then I will only conduct step 3C.
Step 3A. Is there a significant correlation between left variable X in group A and variable Y? (I will conduct Pearson correlation) Step 3B. Is there a significant correlation between right variable X in group A and variable Y? (I will conduct Pearson correlation)
Step 3C. Is there a significant correlation between total variable X in group A and variable Y? (I will conduct a Pearson correlation)
Regards, De
January 2, 2021 at 1:57 pm
I should say that being a budding statistician, this site seems to be pretty reliable. I have few doubts in here. It would be great if you can clarify it:
“A significance level of 0.05 indicates a 5% risk of concluding that a difference exists when there is no actual difference. ”
My understanding : When we say that the significance level is 0.05 then it means we are taking 5% risk to support alternate hypothesis even though there is no difference ?( I think i am not allowed to say Null is true, because null is assumed to be true/ Right)
January 2, 2021 at 6:48 pm
The sentence as I write it is correct. Here’s a simple way to understand it. Imagine you’re conducting a computer simulation where you control the population parameters and have the computer draw random samples from the populations that you define. Now, imagine you draw samples from two populations where the means and standard deviations are equal. You know this for a fact because you set the parameters yourself. Then you conduct a series of 2-sample t-tests.
In this example, you know the null hypothesis is correct. However, thanks to random sampling error, some proportion of the t-tests will have statistically significant results (i.e., false positives or Type I errors). The proportion of false positives will equal your significance level over the long run.
Of course, in real-world experiments, you never know for sure whether the null is true or not. However, given the properties of the hypothesis, you do know what proportion of tests will give you a false positive IF the null is true–and that’s the significance level.
I’m thinking through the wording of how you wrote it and I believe it is equivalent to what I wrote. If there is no difference (the null is true), then you have a 5% chance of incorrectly supporting the alternative. And, again, you’re correct that in the real world you don’t know for sure whether the null is true. But, you can still know the false positive (Type I) error rate. For more information about that property, read my post about how hypothesis tests work .
July 9, 2018 at 11:43 am
I like to use the analogy of a trial. The null hypothesis is that the defendant is innocent. A type I error would be convicting an innocent person and a type II error would be acquitting a guilty one. I like to think that our system makes a type I error very unlikely with the trade off being that a type II error is greater.
July 9, 2018 at 12:03 pm
Hi Doug, I think that is an excellent analogy on multiple levels. As you mention, a trial would set a high bar for the significance level by choosing a very low value for alpha. This helps prevent innocent people from being convicted (Type I error) but does increase the probability of allowing the guilty to go free (Type II error). I often refer to the significant level as a evidentiary standard with this legalistic analogy in mind.
Additionally, in the justice system in the U.S., there is a presumption of innocence and the prosecutor must present sufficient evidence to prove that the defendant is guilty. That’s just like in a hypothesis test where the assumption is that the null hypothesis is true and your sample must contain sufficient evidence to be able to reject the null hypothesis and suggest that the effect exists in the population.
This analogy even works for the similarities behind the phrases “Not guilty” and “Fail to reject the null hypothesis.” In both cases, you aren’t proving innocence or that the null hypothesis is true. When a defendant is “not guilty” it might be that the evidence was insufficient to convince the jury. In a hypothesis test, when you fail to reject the null hypothesis, it’s possible that an effect exists in the population but you have insufficient evidence to detect it. Perhaps the effect exists but the sample size or effect size is too small, or the variability might be too high.
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These days when I look at scientific research papers or review manuscripts, there seems to be almost a competition to have a smaller p value as a means to present more significant findings. For example, a quick Internet search using “ p < 0.0000001” turned up many papers even reporting their p values at this level. Can and should a smaller p value play such a role? In my opinion, it cannot. The current statistical software making possible p value-centered statistical reporting, I believe, is leading scientific inquiry into a quagmire and dead end.
To fully understand why the p value-centered inquiry is the wrong approach, let's firstly understand what p value and hypothesis testing (HT) are and examine how statistical hypothesis testing (SHT) was run prior to the computer era. While p value and HT are both now used under the umbrella of SHT, they had different roots. The p value and its application in scientific inquiry is credited to the English statistician Sir Ronald Aylmer Fisher 1 in 1925. In Fisher's inquiry system, a test statistic is converted to a probability, namely the p value, using the probability distribution of the test statistic under the null hypothesis and the p value was used solely as an aid, after data collection, to assess if the observed statistic is a simply random event or indeed belongs to a unique phenomenon fitting the researchers' scientific hypothesis. 2 Furthermore, 0.05 or 0.01 are not the only p value cutoff scores for the decision. Thus, Fisher's p value inquiry system belongs to a posteriori decision system, which also features, “flexibility, better suited for ad-hoc research projects, sample-based inferential, no power analysis and no alternative hypothesis” (p. 4). 3
HT, on the other hand, was credited to the Polish mathematician Jerzy Neyman and American statistician Egon Pearson 4 in 1933, who sought to improve Fisher's method by proposing a system to apply repetition of experiments. Neyman and Pearson believed that a null hypothesis should not be considered unless one possible alternative was conceivable. In contrast to Fisher's system, Type I error or the error the researchers want to minimize, the corresponding critical region and value of a test must be set up first in the Neyman–Pearson's system, which, therefore, belongs to a priori decision system. In addition, the Neyman–Pearson's system is “more powerful, better suited for repeated sampling projects, deductive, less flexible than Fisher's system and defaults easily to the Fisher's system” (p. 8). 3
The current commonly used SHT is mainly derived from the Neyman–Pearson's system. With the p value conveniently provided by modern statistical software, researchers have started to mix the two systems together and with the result that SHT has started to become a means to foster pseudoscience. 3
A quick review of the SHT practice prior to the computer era may help better explain the above points. A typical SHT can be considered as a decision system by including the following steps:
By going through these steps, you should be able to quickly realize two things: first, SHT is similar to the US criminal court trial system, in which “innocent until proven guilty” is the guiding principle:
If H 0 is rejected when it is true (i.e., Type I error happened), an innocent person may be convicted for a crime they did not commit. Therefore, Type I error in practice is often strictly controlled since the consequences of having a Type I error could be much more serious than a Type II error (failed to convict a criminal). Secondly, before the use of computer software, Type I error or p value had to be determined prior to computing statistics and there were usually only two choices, p = 0.05, which is commonly used in kinesiology research, or p = 0.01, which is commonly used in pharmaceutical research. So, SHT belongs to a priori decision system, i.e., a probability-based evaluation standard or the confidence has to be set up before computing a statistic and making a decision.
An example may be helpful to illustrate the above steps. Say a researcher observed a difference between males and females in body composition and wants to test her research hypothesis that females have a higher percentage of body fat. To do so, she recruited 10 adults (5 females and 5 males) and measured their fat percentage using the underwater weighing method ( Table 1 ).
An example of sex difference on percentage of body fat.
ID | Sex | Fat% |
---|---|---|
1 | Female | 17.55 |
2 | Female | 35.77 |
3 | Female | 29.55 |
4 | Female | 16.84 |
5 | Female | 20.08 |
6 | Male | 20.97 |
7 | Male | 25.59 |
8 | Male | 3.71 |
9 | Male | 5.17 |
10 | Male | 24.27 |
Following the SHT steps, she tested her research hypothesis:
Since there are two groups, she selected the independent t test; given α = 0.05, two-tailed test, and df = 5 ( n of male group) + 5 ( n of female group) − 2 = 8, the critical value according to t value table is 2.306; such, the decision is set as below:
If −2.306 < t statistic observed < 2.306, do not reject H 0 ; If t statistic observed ≤ −2.306 or if t statistic observed ≥ 2.306, reject H 0 .
Female fat%: M = 23.958, SD = 8.330
Male fat%: M = 15.942, SD = 10.646
H 0 was NOT rejected since the observed t statistic is larger than −2.306 and smaller than 2.306.
With convenient and powerful statistical software now available, an extra piece of information is generated when the statistic is computed, i.e., the exact p value along with a specific statistic condition of the sample size and the direction of the test. For the example, for the above research data, if we run the t test using a statistical software, we also get a specific p value corresponding to the t statistic of 1.33, which is p = 0.221. Since it was larger than p = 0.05, one may normally conclude that since H 0 was not rejected, there is no significant difference between males and females in fat percentage. As a result of this additional information, you can see that researchers start to report these specific p values in their research reports and omit other related important information (e.g., the statistics themselves, df , etc .), especially if they have one less than 0.05 or 0.01, which has resulted in the “ p value competition”.
What is the issue with this approach if the p value itself could reach a similar conclusion without other information (e.g., the statistics themselves, df , etc. )? Unfortunately, there are two problems related to this p value only practice. Firstly, it changed the priori nature of the SHT decision deriving, i.e., a Type I error should be selected before one can make a decision. As mentioned above, only two p values, 0.05, which corresponds to a 95% confidence for the decision made or 0.01, which corresponds a 99% confidence, were used before the advent of the computer software in setting a Type I error. Secondly, and a more serious problem, the p value could be impacted by the sample size employed, making it an inconsistent standard in decision-making.
Let's go back to our example to illustrate why p value is not a consistent standard. By looking at the fat percentage means of males and females, you may quickly realize that the difference between the two means is rather large. How was the p value larger than 0.05 when there seems to be an obvious difference between the two means? To get a less-than 0.05 p value or to reject the null hypothesis is, in fact, not difficult as long as we have a large enough statistical power, which is the probability of rejecting the null hypothesis when it is false (i.e., detecting a real difference). There are four factors that may impact the statistical power: (a) α level, (b) one-tailed or two-tailed test, (c) effect size (ES), and (d) sample size. Since the α level (0.05 or 0.01) or the direction of the test (we use a two-tailed test the majority of the time) are often fixed, two things that can affect the statistical power in practice are ES or sample size. For the ES of our example, we computed Cohen's d index: 5
According to the Cohen's ES standard (≥0.8 = large; <0.8 to > 0.2 = medium; ≤0.2 = small), ES of between male and female mean difference in our example indeed belongs to “large”. Thus, the reason the H 0 was not rejected is likely due to the small sample size ( n = 5 for each group) employed. To verify this finding, we compute the sample size needed to get enough power by entering ES of 0.839, the desired statistical power of 0.8 and α level of 0.05 into an online sample size calculator for t test ( http://www.danielsoper.com/statcalc3/calc.aspx?id=47 ). For a two-tailed hypothesis, the recommended sample size per group is 24. For the purpose of illustration, rather than to collect another 19 data points for each group, we simply copied and pasted the existing data three times, which made the sample size of each group 20 and recalculated means, SDs and t test. Here are the results:
Female fat%: M = 23.958, SD = 7.644
Male fat%: M = 15.942, SD = 9.770
p value = 0.006.
As expected, the means remained the same, SDs became slightly smaller, t statistic became larger, and the most important change, of course, is that p value is now less than 0.05 so that the earlier “no difference” conclusion suddenly changed to a “significant” difference. It should be pointed out the p value problem is not only in the situation where a true difference could not be detected when a small sample was employed, but also a little, meaningless difference or no/low correlation could become “significant” when a large sample was employed. 6 It is this inconsistency that makes the p value useless in decision-making.
The above procedures also demonstrated that as long as ES is determined, needed sample size to get a less than 0.05 p value can be easily estimated. Since an absolute evaluation system has been developed for ES (e.g., the small-medium-large rating for Cohen's d ), there is no need to use an extra inconsistent decision-making system. Criticism of the p value and the SHT is not new; in fact, it has a rich history of more than 80 years. 6 , 7 , 8 , 9 The problem of the abuse of the p value, which is often incorrectly used as a symbol of a significant finding, is clearly getting worse due mainly to the exact p values provided by modern statistical software. It is my strong opinion that this reporting practice be stopped. In addition to using ES 5 as an alternative, other recommendations of alternative approaches include exploratory data analysis, 10 confidence interval, 11 meta-analysis, 12 , 13 and Bayesian applications, 14 etc.
Considering p value is currently required by the most journals in the submission process and expected by peer-reviewers, a more practical recommendation to report statistics and p value is as follows:
In summary, due to the conveniently available exact p values provided by modern statistical data analysis software, there is a wave of p value abuse in scientific inquiry by considering a p < 0.05 or 0.01 result as automatically being significant findings and that a smaller p value represents a more significant impact. After explaining the roots of the problem and why p value should not be used in this way, some practical recommendations on appropriately reporting statistical findings, including p value, are provided.
The author declares no competing financial interests.
Peer review under responsibility of Shanghai University of Sport.
A researcher in planning research will develop a hypothesis from the research that will be conducted. The hypothesis is created as a proposition about the population parameters to be tested statistically through samples taken from the population.
Therefore, to determine the conclusion of a study, it is necessary to do statistical hypothesis testing. The statistical hypothesis consists of the null hypothesis (Ho) and the alternative hypothesis (Ha). The null hypothesis is a neutral statement with the sign “=”, while the alternative hypothesis is the opposite of the null hypothesis. Thus, if the null hypothesis is rejected in statistical hypothesis testing, then the alternative hypothesis is accepted and vice versa.
The statistical hypothesis tests the null hypothesis. The determination of whether the null hypothesis is accepted or rejected is carried out based on the purpose of testing the hypothesis. The criteria for accepting or rejecting the null hypothesis are based on the p-value. To determine whether the null hypothesis is accepted or rejected, it refers to the alpha value set in the study.
Generally, the researcher determines the alpha at 1% and 5% for experimental research, while for survey research, it can be set with an alpha value of up to 10%. Because of the importance of knowledge about hypothesis testing based on alpha values, on this occasion, Kanda Data will write about How to Distinguish 0.01, 0.05, and 0.10 Significance Levels in Statistics.
Before the researcher conducts the research, it will be preceded by formulating the null hypothesis and alternative hypotheses. In the formulation of hypotheses, it is known that there are one-tailed hypotheses and two-tailed hypotheses. The writing of these two types of hypotheses is different from one another.
Researchers need to write mathematical equations to facilitate the writing of statistical hypotheses. The one-tailed hypothesis in the mathematical equation contains signs “> and <.” On the other hand, there are signs for the two-tailed hypothesis: “= and ≠. Researchers can choose to use a one-tailed or two-tailed hypothesis according to the research objectives.
Researchers should use a two-tailed hypothesis test for research purposes that are not known for certain whether the effect is negative or positive. Meanwhile, if the researcher can ascertain the direction of the effect, the researcher can choose the one-tailed hypothesis.
To make it easier to understand writing statistical hypotheses, I will give an example of a case study. Suppose a researcher is conducting an observation to determine the difference in rice production before and after introducing new technology.
Researchers observed the average value of rice production before the introduction and then introduced the new technology for six months. Researchers observed the average rice production again after the introduction of new technology was completed. Based on this case example, researchers do not yet know whether the direction is positive or negative. Therefore, the researcher decided to use a two-tailed hypothesis.
Writing a mathematical hypothesis can be written as follows:
Ho: µ = µ 0 there is no significant difference between rice production before the introduction of technology and rice production after the introduction of new technology.
H1: µ ≠ µ 0 there is a significant difference between rice production before the introduction of technology and rice production after the introduction of new technology.
The writing of other hypotheses is adjusted to the researcher’s chosen analysis tool. Although, in principle, the same between the test of influence, correlation, and test of difference, it is better to use a different notation.
Based on what I wrote in the previous paragraph, the researcher can determine the alpha at 1%, 5% or 10%. If the researcher determines an alpha of 5%, it can be analogous that out of 100 trials, failures are less than or equal to 5 times, then the study is declared a success. The same thing can also be analogized if the alpha is determined by 10%, meaning that if the success is 90 times out of 100 trials, the research is declared successful.
In studies with an environment that we can control well, researchers may consider using 5% or 1%. Alpha 5% and 1% can be applied to experimental studies with a relatively controllable research environment.
Especially for research in the medical field, it would be better if the alpha was determined to be smaller, for example, by 1%. On the other hand, survey studies with relatively difficult-to-control environments can determine an alpha of 10%. Therefore, researchers can choose to determine alpha according to their respective fields of study. The smaller the alpha value indicates, the higher the confidence level in a study.
After you understand the difference in alpha levels used in the study, you must have an in-depth understanding of the basic criteria for acceptance of the hypothesis. Following what I wrote in the previous paragraph, what is being tested is the null hypothesis.
Suppose we use the example case that I have written in the paragraph above for the test criteria, namely:
if the p-value > 0.05, then the null hypothesis is accepted
if the p-value ≤ 0.05, then the null hypothesis is rejected (the alternative hypothesis is accepted)
For example, based on the results of the t-test, if the p-value is 0.015, it indicates that the p-value is <0.05. Therefore, based on the criteria for acceptance of the hypothesis, it was decided that Ho was rejected. Because Ho is rejected, we accept the alternative hypothesis, namely that there is a significant difference between rice production before and rice production after introducing new technology.
However, if the alpha determined by the researcher is 1%, the null hypothesis is rejected because the p-value is > 0.01. Researchers can choose to test the hypothesis acceptance criteria following the previously determined alpha. That’s all I can write for all of you. Hopefully, it’s useful. Wait for the update of the Kanda data article next week!
Handling non-normally distributed data by removing outliers, the differences between nominal data scale and ordinal data scale in research variable measurement.
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The significance level is the probability of rejecting the null hypothesis when it is true. Commonly used significance levels are 0.01, 0.05, and 0.10. Remember, rejecting the null hypothesis doesn't prove the alternative hypothesis; it just suggests that the alternative hypothesis may be plausible given the observed data.
The null hypothesis (H0): μ = 2 ounces. The alternative hypothesis: (HA): μ ≠ 2 ounces. The auditor conducts a hypothesis test for the mean and ends up with a p-value of 0.0046. Since the p-value of 0.0046 is less than the significance level of 0.01, the auditor rejects the null hypothesis. He concludes that there is sufficient evidence to ...
The p value is a number, calculated from a statistical test, that describes how likely you are to have found a particular set of observations if the null hypothesis were true. P values are used in hypothesis testing to help decide whether to reject the null hypothesis. The smaller the p value, the more likely you are to reject the null hypothesis.
Using P values and Significance Levels Together. If your P value is less than or equal to your alpha level, reject the null hypothesis. The P value results are consistent with our graphical representation. The P value of 0.03112 is significant at the alpha level of 0.05 but not 0.01.
If the p-value of a hypothesis test is sufficiently low, we can reject the null hypothesis. Specifically, when we conduct a hypothesis test, we must choose a significance level at the outset. Common choices for significance levels are 0.01, 0.05, and 0.10. If the p-values is less than our significance level, then we can reject the null hypothesis.
When we use z z -scores in this way, the obtained value of z z (sometimes called z z -obtained) is something known as a test statistic, which is simply an inferential statistic used to test a null hypothesis. The formula for our z z -statistic has not changed: z = X¯¯¯¯ − μ σ¯/ n−−√ (7.5.1) (7.5.1) z = X ¯ − μ σ ¯ / n.
The P value of 0.03112 is statistically significant at an alpha level of 0.05, but not at the 0.01 level. If we stick to a significance level of 0.05, we can conclude that the average energy cost for the population is greater than 260. A common mistake is to interpret the P-value as the probability that the null hypothesis is true.
If p-value ≥ α, then you don't have enough evidence to reject the null hypothesis. Obviously, the fate of the null hypothesis depends on α. For instance, if the p-value was 0.03, we would reject the null hypothesis at a significance level of 0.05, but not at a level of 0.01. That's why the significance level should be stated in advance and ...
Table of contents. Step 1: State your null and alternate hypothesis. Step 2: Collect data. Step 3: Perform a statistical test. Step 4: Decide whether to reject or fail to reject your null hypothesis. Step 5: Present your findings. Other interesting articles. Frequently asked questions about hypothesis testing.
Testing Hypotheses using Confidence Intervals. We can start the evaluation of the hypothesis setup by comparing 2006 and 2012 run times using a point estimate from the 2012 sample: ˉx12 = 95.61 minutes. This estimate suggests the average time is actually longer than the 2006 time, 93.29 minutes.
Onward! We use p -values to make conclusions in significance testing. More specifically, we compare the p -value to a significance level α to make conclusions about our hypotheses. If the p -value is lower than the significance level we chose, then we reject the null hypothesis H 0 in favor of the alternative hypothesis H a .
The P -value is, therefore, the area under a tn - 1 = t14 curve to the left of -2.5 and to the right of 2.5. It can be shown using statistical software that the P -value is 0.0127 + 0.0127, or 0.0254. The graph depicts this visually. Note that the P -value for a two-tailed test is always two times the P -value for either of the one-tailed tests.
P Values The P value, or calculated probability, is the probability of finding the observed, or more extreme, results when the null hypothesis (H 0) of a study question is true - the definition of 'extreme' depends on how the hypothesis is being tested. P is also described in terms of rejecting H 0 when it is actually true, however, it is not a direct probability of this state.
Let's return finally to the question of whether we reject or fail to reject the null hypothesis. If our statistical analysis shows that the significance level is below the cut-off value we have set (e.g., either 0.05 or 0.01), we reject the null hypothesis and accept the alternative hypothesis. Alternatively, if the significance level is above ...
In null-hypothesis significance testing, the -value is the probability of obtaining test results at least as extreme as the result actually observed, under the assumption that the null hypothesis is correct. A very small p-value means that such an extreme observed outcome would be very unlikely under the null hypothesis. Even though reporting p-values of statistical tests is common practice in ...
A more accurate null hypothesis significance test also has a higher power, because a higher power (1 − β) means a smaller β. The power of a is higher than that of b in Figure 1, indicating that the null hypothesis significance test is more accurate. Determining the power in this way is useful for evaluating the accuracy of the test.
Example 8.4.7. Joon believes that 50% of first-time brides in the United States are younger than their grooms. She performs a hypothesis test to determine if the percentage is the same or different from 50%. Joon samples 100 first-time brides and 53 reply that they are younger than their grooms.
In hypothesis testing, the level of significance is a measure of how confident you can be about rejecting the null hypothesis. This blog post will explore what hypothesis testing is and why understanding significance levels are important for your data science projects. In addition, you will also get to test your knowledge of level of significance towards the end of the blog with the help of quiz.
Example: Criminal Trial Analogy. First, state 2 hypotheses, the null hypothesis ("H 0 ") and the alternative hypothesis ("H A "). H 0: Defendant is not guilty.; H A: Defendant is guilty.; Usually the H 0 is a statement of "no effect", or "no change", or "chance only" about a population parameter.. While the H A, depending on the situation, is that there is a difference ...
Aug 5, 2022. 6. Photo by Andrew George on Unsplash. Student's t-tests are commonly used in inferential statistics for testing a hypothesis on the basis of a difference between sample means. However, people often misinterpret the results of t-tests, which leads to false research findings and a lack of reproducibility of studies.
When the significance level is 0.05 and the null hypothesis is true, there is a 5% chance that the test will reject the null hypothesis incorrectly. If you set alpha to 0.01, there is a 1% of a false positive. If 5% is good, then 1% seems even better, right? As you'll see, there is a tradeoff between Type I and Type II errors.
In summary, due to the conveniently available exact p values provided by modern statistical data analysis software, there is a wave of p value abuse in scientific inquiry by considering a p < 0.05 or 0.01 result as automatically being significant findings and that a smaller p value represents a more significant impact. After explaining the roots of the problem and why p value should not be ...
Generally, the researcher determines the alpha at 1% and 5% for experimental research, while for survey research, it can be set with an alpha value of up to 10%. Because of the importance of knowledge about hypothesis testing based on alpha values, on this occasion, Kanda Data will write about How to Distinguish 0.01, 0.05, and 0.10 ...