Power of test , type 1 - 2 errors

Answer 1

A Type II Error occurs when we fail to reject a false null hypothesis, and the Power of a test is the probability of avoiding a Type II Error.

In this hypothetical situation, we know that the correct decision is to reject the null hypothesis.

We will make a type II error if we fail to reject the null. This will happen when the sample mean is greater than 24.5. The probability of a Type II error (Beta or β) is equal to the probability that x>24.5.

Given the true mean is 23.85, n=8, and variance=.619,

z = (24.5-23.85)/sqrt(0.619/8)=2.34

β = P(x>24.5)=P(z>2.34) = .0.0096. The power of the test is 1-β = .9904

A Type 1 error occurs when we reject a true null. Since the null is false, we are not at risk of a type I error.