6.. In a recent survey conducted by Pew Research, it was found that 156 of 295 a

Question

6.. In a recent survey conducted by Pew Research, it was found that 156 of 295 adult Americans without a high school diploma were worried about having enough saved for retirement. Does the sample evidence suggest that a majority of adult Americans without a high school diploma are worried about having enough saved for retirement? Answer the following:

a. What would it mean to make a Type II error for this test?

b. If the researcher decides to test this hypothesis at the 0.05 level of significance, determine the probability of making a Type II error if the true population proportion is 0.54. What is the power of the test? Please can you number the answers back to me. Please I will appreacite if you can number the answers if me, Thank you.

Explanation / Answer

Sol:

Part a:

We know that the type II error is defined as the probability of do not rejecting null hypothesis H0 when it is not true. For the given scenario, the type II error is defined as the probability of do not rejecting the claim that not a majority of adult Americans without a high school diploma are worried about having enough saved for retirement; when actually majority of adult Americans without a high school diploma are worried about having enough saved for retirement.

Part b:

We are given

X = 156

n = 295

Sample proportion = P = 156/295 = 0.528814

Population proportion = p = 0.54, so q = 1 – p = 1 – 0.54 = 0.46

Z = (P – p) / sqrt(pq/n) = (0.528814 - 0.54) / sqrt(0.54*0.46/295) = -0.38549

P(Z<-0.38549) = 0.349938

For p = 0.54, z = 0.00, P(Z<0.00) = 0.5000

Type II error = ? = 0.5000 - 0.349938 = 0.150062

Power = 1 – ? = 1 - 0.150062 = 0.849938

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