Refer again to Fig. 10.8 on page 427, where each pair of graphs shows the distributions of a variable on two populations. Suppose that, in each case, you want to perform a small-sample hypothesis test based on independent simple random samples to compare the means of the two populations. In each case, decide which, if any, of the following tests is preferable: the pooled $t$-test, the non pooled $t$-test, or the Mann-Whitney test. Explain your answers.

Given in the question that, refer figure 10.8

We have to decide which, if any, of the following tests is preferable: the pooled $t$-test, the non pooled $t$-test, or the Mann-Whitney test. Explain your answers.

Consider the given figure.

The two standard deviation values are close in the given figure (a), and the two distributions are normal. As a result, pooled testing is preferred.

Consider the given figure (b)

The two standard deviation values are not similar in figure 10.8 (b), and the two distributions are not normal. As a result, non pooled testing is favored.

Consider the given figure (c)

Figure 10.8 (c) shows that the two distributions are not normal and do not have the same shape. The Mann-Whitney test method assumption is met. Thus, the Man-Whitney test is recommended.

Given in the question that,

The two distributions are not normal and do not have the same shape, as seen in figure 10.8 (d). The Mann-Whitney test procedure's assumption is not met. As a result, the Man-Whitney test is not recommended.