7.9 Carrying Out a Test for the Difference of Two Population Means

To carry out a two-sample t-test for the difference of two population means, calculate the t statistic by dividing the difference in sample means by the standard error, find the degrees of freedom with technology, get the p-value, and compare it to your significance level. If the p-value is at or below alpha, reject the null hypothesis and state your conclusion in context.

Why This Matters for the AP Statistics Exam

This topic is the calculation and conclusion stage of comparing two means, which shows up often in free-response questions that ask whether data give convincing evidence of a difference. When a question asks for convincing evidence, it is asking for a significance test, not just a description of the numbers. You need to identify the correct parameter and hypotheses, check conditions, calculate the test statistic and p-value, and then write a conclusion that links the p-value to the decision in context. Being precise with notation and showing your work clearly is important for strong exam responses on both multiple-choice and free-response questions.

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Key Takeaways

• The test statistic is t = ((x̄₁-x̄₂)-(μ₁-μ₂))/√(s₁²/n₁+s₂²/n₂), and under the null the (μ₁-μ₂) term is 0.
• The standard error of the difference is √(s₁²/n₁+s₂²/n₂).
• Degrees of freedom fall between the smaller of n1-1 and n2-1 and n1+n2-2; technology gives a precise value.
• The p-value is computed by assuming the null is true, meaning the two population means are equal.
• Compare the p-value to alpha: if p ≤ alpha, reject H0; if p > alpha, fail to reject H0.
• Write down the t statistic, degrees of freedom, and p-value, and state your conclusion in context.

Calculating the Test Statistic

Once you have confirmed the conditions for a two-sample t-test are met, you can calculate the test statistic and p-value to decide whether the difference between the two means is statistically significant.

You are comparing one quantitative variable across two independent samples. The first step is finding the difference between the two sample means and dividing it by the standard error of the difference.

Degrees of Freedom

• Calculating by hand: take the smaller of the two sample sizes and subtract 1. This is the conservative approach, similar to what you did with a single sample in Unit 7.5.
• Using technology such as a graphing calculator: the degrees of freedom come with the output, and the value falls between the smaller of n1-1 and n2-1 and n1+n2-2.

t Statistic

The general test statistic formula is:

$\frac{observed-expected}{\sigma}$

For the difference of two population means, this becomes:

$\frac{\bar{x}_{1}-\bar{x}_{2}}{\sqrt{\frac{{s^2}_{1}}{n_1}+\frac{{s^2}_{2}}{n_2}}}$

The full form includes the hypothesized difference (μ₁-μ₂) in the numerator, but since the null hypothesis usually sets that difference to 0, the term drops out. You can build this from the general test statistic formula and the standard error formulas on the Formula Sheet, so you do not need to memorize it.

Calculating the P-Value

With your degrees of freedom and t statistic, you can use the table on the Formula Sheet. Find the row for your degrees of freedom, look across to find the t value closest to yours, and use the tail probability that matches.

A more exact approach is to run a two-sample t-test using technology such as a graphing calculator. You can either type in the summary statistics or enter the raw data into a list. The output gives you the t statistic, degrees of freedom, and p-value.

Remember that the p-value is computed by assuming the null hypothesis is true, which means assuming the two population means are equal.

On a free-response question, write down the t statistic, the degrees of freedom, and the p-value so your work is complete and clear.

Making the Decision and Stating a Conclusion

Once you have your p-value, compare it to the significance level (often 0.05) to evaluate the null hypothesis.

• If the p-value is at or below alpha, reject H0. You have convincing evidence for the alternative hypothesis.
• If the p-value is greater than alpha, fail to reject H0. You do not have convincing evidence for the alternative.

Always state your conclusion in context. For a study comparing the mean number of green beans picked from two fields, a conclusion might read:

Since the p-value is essentially 0 and less than 0.05, we reject H0. We have convincing evidence that the true mean number of green beans picked from Field A differs from the true mean picked from Field B.

That conclusion works because it compares the p-value to the significance level, states the decision about H0, connects to the alternative hypothesis, and stays in the context of the problem.

Worked Example: Comparing Recovery Times

Here is how the pieces fit together in a real comparison. In a study comparing mean recovery times for two surgical procedures to repair a torn ACL, one group had a sample size of 110 and the other had 100. The degrees of freedom fall between 100 (the smaller of 110 and 100) and 208 (110 + 100 − 2). Using technology, the degrees of freedom came out to about 207.18. With a test statistic of t ≈ 7.13, the p-value is the area greater than 7.13 for a t-distribution with df = 207.18. That very large t statistic gives a tiny p-value, which would lead you to reject the null and conclude there is convincing evidence of a difference in mean recovery times.

How to Use This on the AP Statistics Exam

Free Response

• State hypotheses in terms of population parameters: H₀: μ₁-μ₂=0 (or μ₁=μ₂) and an alternative that matches the question.
• Name the procedure (two-sample t-test for a difference of means) and check conditions before calculating.
• Show the test statistic, degrees of freedom, and p-value.
• Compare the p-value to alpha with a numerical reference, such as "Because p < 0.05, we reject H0."
• End with a conclusion in context that connects back to the alternative hypothesis.

MCQ

• Be ready to identify the correct standard error, √(s₁²/n₁+s₂²/n₂), and the correct test statistic.
• Know that under the null, the (μ₁-μ₂) term equals 0.
• Recognize that degrees of freedom from technology fall between the smaller of n1-1 and n2-1 and n1+n2-2.

Common Trap

• "Convincing evidence" signals a significance test, not just a description of the data.
• Rejecting H0 is not the same as proving the alternative; it means the data are unlikely under the null.

Common Misconceptions

• The two-sample t-test uses two independent samples. Do not confuse it with a matched-pairs setup, where you analyze differences as a single sample.
• The p-value is not the probability that the null hypothesis is true. It is computed assuming the null is true.
• Failing to reject H0 does not prove the means are equal. It only means you lacked convincing evidence of a difference.
• The hypotheses must be written with population parameters (μ₁ and μ₂), not sample statistics.
• A formal decision compares the p-value to alpha directly. Vague statements like "the p-value is small" without a comparison are not enough.
• Degrees of freedom for two samples are not simply n1-1. By hand you use the smaller sample size minus 1 as a conservative value, while technology gives a more exact number.

Related AP Statistics Guides

• Unit 7 Overview: Means
• 7.2 Constructing a Confidence Interval for a Population Mean
• 7.1 Introducing Statistics: Should I Worry About Error?
• 7.4 Setting Up a Test for a Population Mean
• 7.3 Justifying a Claim About a Population Mean Based on a Confidence Interval
• 7.6 Confidence Intervals for the Difference of Two Means