e-Statistics

Guide to Worksheet 7-8

Worksheet No.7. A short report in Problem 4 and 5 must include:

Introduction of the study (the description of data, and summary statistics);
a formal statistical analysis (hypotheses to set up, the choice of test and the discussion on its appropriateness);
the result and your conclusion.

All the data assume two independent groups, and either general or pooled t-test should be performed.

When the sample variances are not considered to be equal, the general t-test should be used [Problem 1,2,5].
If the sample variances can be considered to be about equal, the use of pooled t-test should be recommended [Problem 3, 4]. The conclusion from both general and pooled t-test are often the same [Problem 3, 4; both the tests produce very small p-values]. However, different tests could provide different conclusions; this is why it is important to choose the most appropriate test.
The two-sided test ( $H_A: \mu_1 \neq \mu_2$ ) must be used whenever the use of two-sided test is explicitly indicated [2,3].
Although the one-sided test ( $H_A: \mu_1 < \mu_2$ or $H_A: \mu_1 > \mu_2$ ) is indicated [1,4,5], the two-sided test could be used instead. This may end up with more conservative result (i.e., much harder to reject ). After you reject , you should discuss whether $\mu_1 < \mu_2$ or $\mu_1 > \mu_2$ in a report by using the confidence interval for the difference $\mu_1 - \mu_2$ [4,5].

Worksheet No.8. The t-test requires the ``normality assumption.'' That is, either (a) the sample size of each group (or, of the difference if paired) is at least 30, or (b) the sample distribution is approximately normal. In order to justify (b), you may use QQ-plot, and see whether it looks reasonably straight. Wilcoxon tests assume that the shapes and the spreads (i.e., the sample standard deviations) of the two sample distributions are reasonably close for the rank-sum test, or that the sample distribution of the difference is symmetric for the signed-rank test. While these tests, t-test and nonparametric, perform well under moderate violations of their respective assumptions, none of these tests are universally reliable.

Sample sizes are small and QQ plots do not look favorable to normality in Problem 1. Thus, Wilcoxon rank sum test (p-value: 0.13) may be the most appropriate test.
In Problem 2, sample sizes are modest, but QQ plots do not necessarily indicate normality as well. On the other hand the sample variances are quite different, which does not fit well for Wilcoxon test. Thus, neither general t-test (p-value: 0.001) nor Wilcoxon rank sum test (p-value: 0.004) is exactly the best choice, and both tests should be tried.
Data are paired in Problem 3. Sample size is small and QQ plot for the difference suggests a possible outlier. Both paired t-test (p-value: 0.003) and Wilcoxon signed-rank test (p-value: 0.031) should be performed.
Data are paired in Problem 4. Sample size is small to modest, and QQ plot for the difference suggests a normal distribution except for one extreme value. Paired t-test (p-value: 0.206) may be good enough, but Wilcoxon signed rank test (p-value: 0.138) can be considered as well.
Data are paired in Problem 5. Sample size is modest and QQ plot for the difference suggests a normal distribution except for one extreme value. Paired t-test (p-value: 0.011) may be good enough, but Wilcoxon signed rank test (p-value: 0.012) can be considered as well.