Question

Power questions. Suppose a population follows a normal distribution with a standard deviation of 1. We take a random sample of n= 100 individuals from this population. We know, then, that sample mean for our population will have the property that X∼N(μ,1/100).

(a)If μ= 10, find the probability that X >10.1645

(b)If μ= 10.3, find the probability that ̄X >10.1645

(c)Now, consider testing H₀:μ= 10 vs H₁:μ >10, using α= 0.05. What is the rejection rule for this test? Give me the rejection rule in two forms:
Reject H₀ if Z > ___________
Reject H₀ if X > ___________

(d) Considering the rejection rule you came up with in the previous part of the problem, what is the probability that you will end up rejecting H₀ if μ= 10? This is the significance level of the test (the probability you will reject H₀ when H₀ is true).

(e) Considering the rejection rule you came up with in the previous part of the problem, what is the probability that you will end up rejecting H₀ if μ= 10.3?This is the power of the test when μ= 10.3 (the probability you will reject H₀ when H₀ is false, and μ= 10.3).

(f) What is the power of this test when μ= 10.5?

Answer 1