3) A study of estrogen levels in two different groups of women finds the following results (in pg/mL): Group A: 18.7 20.6 20.7 19.7 19.9 19.4 20.2 21.6 18.8 14.1 21.6 16.2 21.7 20.8 19.3 21.3 19.9 20.8 23.2 Group B: 15.2 36.2 27.5 4.7 24.5 29.4 25.9 62.8 Some summary statistics to help you: n Group A: Group B: 19.921 2.0457 19 28.275 16.9169 8 Is there a difference in estrogen levels? (Note: d.f. = 7.0864 for Welch's t-test). Use = 0.05. Make sure you give Ho and H (symbols are okay), and clearly write out your conclusion. 4) Repeat (3), but this time assume equal variances (i.e., use the classic t-test). Use the same level of a you used before. Make sure you give Ho and H (symbols are okay), and clearly write out your conclusion. 5) Now let's compare the tests from problems (3) and (4) (a) Which test (problem (3) or problem (4)) lets you reject the null hypothesis? (b) Which test do you *think* has more power? Usually, but not always(!!), the test with the most power has a lower p-value. (c) If you don't know that the population variances are equal, which test should you use? (d) which test should you use here? (Do you know if the population variances are equal?) Big hint and comment: This is an example of when the classic (= equal variance) t-test can make a serious mistake. Rejecting a null hypothesis when it is not appropriate is a pretty serious mistake. Note also that the sample sizes are very different. (If you're theoretically inclined, here's what happens: even though we set a = 0.05, the classic t-test makes a type I error at a much higher rate than 5%. In other words, despite setting a = 0.05, the actual value of a > 0.05, which is obviously not good!)