Question

A null hypothesis is not rejected at a given level of significance. As the assumed value of the mean gets further away from the true population mean, the Type II error will

Answer 1

ANSWER:-

Basic Concepts

as the mean gets farther from the true mean the power of the test increases and hence the type 2 error decrease.

as type 2 error( β) =acceptance of false hypothesis

and power of the test( 1-β) = rejection the false hypothesis.

type 2 error +power of the test =1

so as power of test increases the type 2 error decreases.

E.g.

Let X denote the IQ of a randomly selected adult American. Assume, a bit unrealistically, that X is normally distributed with unknown mean μ and standard deviation 16. Take a random sample of n = 16 students, so that, after setting the probability of committing a Type I error at α = 0.05, we can test the null hypothesis H₀: μ = 100 against the alternative hypothesis that H_A: μ > 100.

What is the power of the hypothesis test if the true population mean were μ = 108?

Solution:-

Setting α, the probability of committing a Type I error, to 0.05, implies that we should reject the null hypothesis when the test statistic Z ≥ 1.645, or equivalently, when the observed sample mean is 106.58 or greater:

because we transform the test statistic Z to the sample mean by way of:

Now

and illustrated here:

In summary, we have determined that we have (only) a 64.06% chance of rejecting the null hypothesis H₀: μ = 100 in favor of the alternative hypothesis H_A: μ > 100 if the true unknown population mean is in reality μ = 108.

shifting away from true mean

What is the power of the hypothesis test if the true population mean were μ = 112?

Solution. Because we are setting α, the probability of committing a Type I error, to 0.05, we again reject the null hypothesis when the test statistic Z ≥ 1.645, or equivalently, when the observed sample mean is 106.58 or greater. That means that the probability of rejecting the null hypothesis, when μ = 112 is 0.9131 as calculated here:

and illustrated here:

In summary, we have determined that we now have a 91.31% chance of rejecting the null hypothesis H₀: μ = 100 in favor of the alternative hypothesis H_A: μ > 100 if the true unknown population mean is in reality μ = 112. Hmm.... it should make sense that the probability of rejecting the null hypothesis is larger for values of the mean, such as 112, that are far away from the assumed mean under the null hypothesis.

Again shifting the from true mean

What is the power of the hypothesis test if the true population mean were μ = 116?

Solution. Again, because we are setting α, the probability of committing a Type I error, to 0.05, we reject the null hypothesis when the test statistic Z ≥ 1.645, or equivalently, when the observed sample mean is 106.58 or greater. That means that the probability of rejecting the null hypothesis, when μ = 116 is 0.9909 as calculated here:

and illustrated here:

In summary, we have determined that, in this case, we have a 99.09% chance of rejecting the null hypothesis H₀: μ = 100 in favor of the alternative hypothesis H_A: μ > 100 if the true unknown population mean is in reality μ = 116. The probability of rejecting the null hypothesis is the largest yet of those we calculated, because the mean, 116, is the farthest away from the assumed mean under the null hypothesis.

OBSERVATION:-

From the above two cases as the assumed mean shifted away from the true mean the power of test increased that's why type 2 error decreased

**please comment if there is any doubt in understanding and please like it.