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In: Statistics and Probability

Power questions. Suppose a population follows a normal distribution with a standard deviation of 1. We...

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?

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