SECTION 8.2 Testing the Difference Between Means (Independent Samples, s1 and s2 Unknown) 429 The requirements for the z@test described in Section 8.1 and the t@test described in this section are shown in the flowchart below. Two-Sample Tests for Independent Samples You cannot use the z-test or the t-test. Are both population standard deviations known? Are both populations normal or are both sample sizes at least 30? Are both populations normal or are both sample sizes at least 30? No Are the population variances equal? Yes Use the z-test. Yes Yes Yes No No No Use the t-test with and d.f. = n1 + n2 − 2. σ x1 −x2 = + ˆ 1 n1 1 n2 Use the t-test with and d.f. = smaller of n1 − 1 and n2 − 1. x1 −x2 = + s1 n1 s2 n2 2 2 s s • Using a Two-Sample t-Test for the Difference Between Means (Independent Samples, S1 and S2 Unknown) In Words In Symbols 1. Verify that s1 and s2 are unknown, the samples are random and independent, and either the populations are normally distributed or both n1 Ú 30 and n2 Ú 30. 2. State the claim mathematically State H0 and Ha. and verbally. Identify the null and alternative hypotheses. 3. Specify the level of significance. Identify a. 4. Determine the degrees of freedom. d.f. = n1 + n2 - 2 or d.f. = smaller of n1 - 1 and n2 - 1 5. Determine the critical value(s). Use Table 5 in Appendix B. 6. Determine the rejection region(s). 7. Find the standardized test statistic t = 1x1 - x22 - 1m1 - m22 sx 1 -x2 and sketch the sampling distribution. 8. Make a decision to reject or fail to If t is in the rejection region, reject the null hypothesis. then reject H0. Otherwise, fail to reject H0. 9. Interpret the decision in the context of the original claim. GUIDELINES Picturing the World A study published by the American Psychological Association in the journal Neuropsychology reported that children with musical training showed better verbal memory than children with no musical training. The study also showed that the longer the musical training, the better the verbal memory. Suppose you tried to duplicate the results as follows. A verbal memory test with a possible 100 points was administered to 90 children. Half had musical training, while the other half had no training and acted as the control group. The 45 children with training had an average score of 83 with a standard deviation of 5.7. The 45 students in the control group had an average score of 80 with a standard deviation of 6.2. (Adapted from American Psychological Association) At A = 0.05, is there enough evidence to support the claim that children with musical training have better verbal memory test scores than those without training? Assume the population variances are equal.
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