Question

A university will add fruit juice vending machines to its classroom buildings if the student body president is convinced that more than 20 percent of the students will use them. A random sample of n students will be selected and asked whether or not they would use the vending machines. A large-sample test for proportions at the significance level of a = 0.05 will be performed. The null hypothesis that the proportion of all students who would use the vending machines is 20 percent will be tested against the alternative that more than 20 percent of all students would use them. For which of the following situations would the power of the test be highest?
A .The sample size is n = 750, and 20 percent of all students actually use the vending machines.
B. The sample size is n = 750, and 50 percent of all students actually use the vending machines.
C. The sample size is n = 1,000, and 55 percent of all students actually use the vending machines.
D. The sample size is n = 1,000, and 25 percent of all students actually use the vending machines.
E. The sample size is n = 500, and 50 percent of all students actually use the vending machines.

Answer 1

Answer:

The power of the test will be highest for;

C. The sample size is n = 1,000, and 55 percent of all students actually use vending machines

Step-by-step explanation:

For a large sample test for proportions, we have;

$z=\dfrac{\hat{p}-p}{\sqrt{\dfrac{p \cdot q}{n}}}$

Where;

$\hat p$ = The percentage of the hypothesized proportion = 20% = 0.20

p = The actual proportion

n = The size of the sample

The power of the test increases when;

The sample size 'n' is large

When the true vale of the measured items in the test is much larger than the value set as the null hypothesis

When the significant level is high

Therefore, we have;

The power of the test will be highest when the sample size is n = 1,000, and 55 percent of all students actually use vending machines