Question

In a recent survey conducted by Pew Research, it was found that 159 of 295 adult Americans without a high school diploma were worried about having enough saved for retirement.

1. Does the sample evidence suggest that a majority of adult Americans without a high school diploma are worried about having enough saved for retirement? Use the Classical approach with a 10% level of significance.

2. Based on your results of the hypothesis test—the decision you made in part a, what type of error might have occurred? Please explain.

Answer 1

a.

Let p be the true proportion of adult Americans without a high school diploma are worried about having enough saved for retirement

H0: p = 0.5

H1 : p > 0.5

Observed proportion, $\hat{p}$ = 159 / 295 = 0.539

Standard error of sample proportion, SE = $\sqrt{0.539 * (1 - 0.539)/295}$ = 0.029

Test statistic, z = ($\hat{p}$ - p) / SE = (0.539 - 0.5) / 0.029 =  1.34

For 10% level of significance, the critical z value for one tail test is 1.28

As, observed test statistic is greater than the critical value, we reject H0 and conclude that there is significant evidence that p > 0.5 and majority of adult Americans without a high school diploma are worried about having enough saved for retirement.

b.

As, we have rejected the null hypothesis in part (a), we may have made type I error. In case, the null hypothesis is true, we have made type 1 error as we have rejected null hypothesis.