Hypothesis testing with confidence interval

Question

In this question, I have no idea why there shows the value 375, how can I use it? And also for part(iii) I cannot get the idea what the question is asking. Could anyone helps me out?

Answer 1

Without the context of this question in the book or course, there are several ways to interpret this question. I will pick the one that seems most likely to me.

You are given information for finding the sample variances $S_x^2 = 285/9$ and $S_y^2 = 375/14.$

In (i) you are asked to test $H_0: \mu_x \ge \mu_y$ against $H_a: \mu_x < \mu_y$ at the 5% level. This is a one-sided, two-sample test. For normal data, two such tests are in common use: the 'pooled' test (in which it is assumed that $\sigma_x^2 = \sigma_y^2$ and the 'Welch' or 'separate variances' test (in which the assumption of equal population variances is not made).

I suppose (i) intends you to assume equal variances and do the pooled test. It would use the following 'pooled' estimate of the 'common' (shared) population variance $\sigma^2 = \sigma_x^2 = \sigma_y^2,$ $$S_p^2 =\frac{(n_x - 1)S_x^2 + (n_y - 1)S_y^2}{n_x + n_y -2} = \frac{285 + 375}{23},$$ to compute the $T$-statistic, having Student's t distribution with $n_x + n_y - 2$ degrees of freedom (under $H_0$). I refer you to your text for details. [By contrast, the Welch test does not use $S_p^2$ and has a formula for determining the degrees of freedom for the test statistic.]

In part (ii), I suppose you are asked to test whether the equal variance assumption is realistic. That is, to test $H_0: \sigma_x^2/\sigma_y^2 = 1$ against the alternative $H_a: \sigma_x^2/\sigma_y^2 \ne 1,$ at the 1% level. This test uses the variance-ratio statistic $F = S_x^2 / S_y^2,$ which has Snedecor's F-distribution with numerator degrees of freedom 9 and denominator degrees of freedom 14 (under $H_0$). Again, I refer you to your text for details.

Note: Based on theory and many simulation studies, I believe that current statistical practice is to use the Welch separate-variances test, unless there is solid prior knowledge or experience that the two population variances must be equal.

The suggestion to do a preliminary F-test to determine whether to do a pooled or Welch test has been deprecated. One of several difficulties lies in trying to assign an overall level of significance to the 'hybrid' two-test procedure. Another difficulty is that the F-test has poor power unless sample sizes are quite large, and using a pooled test with unequal population variances can lead to misleading P-values, especially when sample sizes are unequal.

However, it is a reasonable exercise, in a beginning statistics course, to practice these three two-sample tests, pooled and Welch t, and F. I believe this exercise is asking you to practice the pooled and F tests.