Wiki Page, Math 13 - Perez Sandbox : Module 24 - Wrap Up "Hypothesis Test for a Population Mean"

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2026-01-17T21:15:43-08:00

Let’s Summarize

In this "Hypothesis Test for a Population Mean" module, we looked at the four steps of a hypothesis test as they relate to a claim about a population mean.

Step 1: Determine the hypotheses.

The hypotheses are claims about the population mean, .

The null hypothesis is a hypothesis that the mean equals a specific value, .

The alternative hypothesis is the competing claim that is less than, greater than, or not equal to the .

When is or , the test is a one-tailed test.

When is , the test is a two-tailed test.

In the case of a matched pairs design, the variable is the difference in two measurements and is the mean of the differences in the population.

Step 2: Collect the data.

Since the hypothesis test is based on probability, random selection or assignment is essential in data production. Additionally, we need to check whether the t-model is a good fit for the sampling distribution of sample means. To use the t-model, the variable must be normally distributed in the population or the sample size must be more than 30. In practice, it is often impossible to verify that the variable is normally distributed in the population. If this is the case and the sample size is not more than 30, researchers often use the t-model if the sample is not strongly skewed and does not have outliers.

Step 3: Assess the evidence.

If a t-model is appropriate, determine the t-test statistic for the data’s sample mean.

Use the test statistic, together with the alternative hypothesis, to determine the P-value.

The P-value is the probability of finding a random sample with a mean at least as extreme as our sample mean, assuming that the null hypothesis is true.

As in all hypothesis tests, if the alternative hypothesis is greater than, the P-value is the area to the right of the test statistic. If the alternative hypothesis is less than, the P-value is the area to the left of the test statistic. If the alternative hypothesis is not equal to, the P-value is equal to double the tail area beyond the test statistic.

Step 4: Give the conclusion.

The logic of the hypothesis test is always the same. To state a conclusion about , we compare the P-value to the significance level, .

If P , we reject . We conclude there is significant evidence in favor of .

If P , we fail to reject . We conclude the sample does not provide significant evidence in favor of .

We write the conclusion in the context of the research question. Our conclusion is usually a statement about the alternative hypothesis (we accept or fail to accept ) and should include the P-value.

Other Hypothesis Testing Notes

Remember that the P-value is the probability of seeing a sample mean at least as extreme as the one from the data if the null hypothesis is true. The probability is about the random sample; it is not a “chance” statement about the null or alternative hypothesis.

Hypothesis tests are based on probability, so there is always a chance that the data has led us to make an error.

If our test results in rejecting a null hypothesis that is actually true, then it is called a Type I error.

If our test results in failing to reject a null hypothesis that is actually false, then it is called a Type II error.

If rejecting a null hypothesis would be very expensive, controversial, or dangerous, then we really want to avoid a Type I error. In this case, we would set a strict significance level (a small value of , such as 0.01).

Finally, remember the phrase “garbage in, garbage out.” If the data collection methods are poor, then the results of a hypothesis test are meaningless.

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