Question

I would like the whole Question done on r studio with the R Code.

1. In this question we will evaluate type I and type II error probabilities for one-sided tests. We will consider normally distributed data, with unit variance and independent obervations. We will use Ho : μ-0 for the null and H1 : μ-1 for the alternative, unless otherwise stated. (a) Suppose we have n-6 observationsx. What is the sampling distribution of the (10 marks) sample mean (that is, ofx-(xi+x6)?) (b) We want a test with size a -0.05. This test is to be of the form "reject Ho if the sample (10 marks) mean x exceeds T" (where T is a value to be determined). You will recall that a is the probability of rejecting Ho when true. Find an appropriate value of T (c) Calculate ß, the probability of failing to reject the null hypothesis when the alternative (10 marks) is true, and state the power of the test. (d) Consider a test of size a0.01. Calculate the power of this test. (e) How many observations would it take to have a size of at most 0.01 and a power of at (10 marks) (10 marks) least 0.99? (f) (harder) Now we will consider the case where the null and alternative hypotheses are (20 marks) very close. We will have Ho: H but now H: 0.02. Now how many observations are needed to ensure a is at most 0.01 and the power is at least 0.99?

Answer 1

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