## Description

1. Chapter 4, problem 30

2. Chapter 4, problem 32

3. Chapter 5, problem 19

4. Consider the Bumpus’s data in Chapter 2, compute the power of the two-sided two sample t-test of

size 0.05 (i.e., reject the null hypothesis if the absolute value the t-statistic is greater than or equal to

2), under the alternative that µx − µy = ¯x − y¯ = 0.01 and σ = sp = 0.0214.

5. Show that the two-sided two sample t-test is equivalent to the anova F-test, if the number of groups is

two.

6. Consider X1, …, X10 are i.i.d. N(0, σ2

), Y1, …, Y10 are i.i.d. N(µ, σ2

) and hypothesis testing:

H0 : µ = 0,

HA : µ 6= 0.

Compute the power of a two sided two sample t-test of size 0.05 when σ

2 = 1 and µ = 0.1, 0.5, 1, and

2. Plot the power as a function of µ. Then, increase the sample size in each group to 20 and draw the

power function in the same plot as that of the sample size 10.

7. Under the setting of the previous problem, show that, under the null hypothesis, the p-value follows

the uniform distribution on the interval [0, 1] and perform simulations to confirm it.

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