In this question we will evaluate type I and type II error probabilities for one-sided tests. We will consider normally distributed data, with known unit variance and independent obervations. We will use H0 : = 0 for the null and H1 : = 1 for the alternative, unless otherwise stated.

Suppose we have n = 6 observations x1, . . . , x6. If the null is true, what is the sampling distribution of the sample mean (that is, of x = 1 6 (x1 + + x6)?)

a. We want a test with size = 0.05. This test is to be of the form "reject H0 if the sample mean x exceeds T" (where T is a value to be determined). You will recall that is the probability of rejecting H0 when true. Find an appropriate value of T.

b. Calculate , the probability of failing to reject the null hypothesis when the alternative is true, and state the power of the test.

c. Consider a test of size = 0.01. Define the power of a test, and calculate the power of this test.

d. Now we will consider the case where the null and alternative hypotheses are very close. We will have H0 : = 0 but now H1 : = 0.02. Now how many observations are needed to ensure is at most 0.01 and the power is at least 0.99?

PLS use the R studio to solve it and shou every step and the R code.