4.10 Carrying Out a Test for the Difference Between Two Population Means

Calculate an appropriate test statistic and p-value for testing a hypothesis for the difference between two population means.

• The test statistic for a two-sample t-test for the difference between two population means is ()xx- -t = 12 0 ss22 1 + 2 nn1 2 . The t-statistic has a t-distribution when the null hypothesis is true. The t-statistics with the degrees of freedom can be found using technology. The degrees of freedom fall between n12+-n 2 and the smaller of n1 −1 and n2 −1.
• The p-value for a two-sample t-test for the difference between two population means can be found using the appropriate t-distribution table or from the appropriate t-distribution using technology.

Interpret the p-value of a hypothesis test for the difference between two population means.

• The p-value is the probability of obtaining a test statistic as extreme or more extreme than the test statistic that was observed (i.e., in the direction of the alternative hypothesis) given that the null hypothesis is true. An interpretation of the p-value of a hypothesis test for a two-sample test for the difference between two population means should include a statement that the p-value is computed by assuming that the null hypothesis is true (i.e., by assuming the population means are equal to each other in context).

Justify a claim about the populations based on the results of a hypothesis test for the difference between two population means.

• A formal decision explicitly compares the p-value to the significance level, α. If the p-value ≤α , then reject the null hypothesis, H:01 μμ-= 2 0 or H:01 μμ= 2. If the p-value >α, then fail to reject the null hypothesis.
• The results of a hypothesis test for a twosample t-test for a difference between two population means can serve as the statistical reasoning to support the answer to an investigative question about the two populations that were sampled.
• A conclusion for the hypothesis test for the difference between two population means is stated in context consistent with, and in terms of, the alternative hypothesis using nondefinitive language. The conclusion should contain a reference to the parameters and the populations.