Question

2. (25 points) Suppose that we have a sample of size n 64, we know the population standard deviation is σ 48, and we are considering a normally distributed population, we want to test the hypotheses: Ho : μ-200 Hi 200 We are going to use a z-test because σ is known. We will use a significance level of:-0.05. (a) What is the critica z value? In other words, for what values of z will we reject Ho? (b) What is the critical value in terms of ? In other words, for what values of will we reject Ho? (c)Suppose that, in reality, 208. Find the power of the test. That is, find the probability of rejecting Ho given 1-208. (d) Suppose that, in reality, 216. Find the power of the test. That is, find the probability of rejecting Ho given H216 (e) What is the probability of committing a Type I error if thel hypothesis is true?
  

Use R to find to find the answers to the problems

Answer 1

a) This is a right tailed test.

The critical value can be found by using the command:

alpha=0.05
z_alpha=qnorm(1-alpha) #This gives you the point below which 95% of the area lies, Hence above this point only 5% of the area lies.

This gives the critical value as z-alpha= 1.644854

b) The test statistic is:

X-200 X-200 48 6 V64

The critical value is 1.644854. Equating these gives, from R:

critical_xbar=6*z_alpha+200
critical_xbar

Output:

 209.8691

c)

X-208 209.8691 208 P (Reject Ho 208)-PX> 209.8691]- P 6 6

                                                                                              

In R terms this is 1-pnorm((critical_xbar-208)/6)

Output:

0.3777026

d) The same procedure as above except in R we will find 1-pnorm((critical_xbar-216)/6)

Output:

0.3777026

e) P(Type I error) is given as:

X-200 209.8691-200 P (Reject HoP-200)-PIX > 209.86911-P 6

In R: 1-pnorm((critical_xbar-200)/6)

Output:

0.05