A major videocassette rental chain is considering opening a new store in an area that currently does not have any such stores. The chain will open if there is evidence that more than 5,000 of the 20,000 households in the area are equipped with videocassette recorders (VCRs). It conducts a telephone poll of 300 randomly selected households in the area and finds that 96 have VCRs. At =2.5% and sample data, what would be your recommendation to the rental chain management? a. Since the critical test value is 1.96, we are going to reject the null hypothesis. The rental chain store should open a store in this area.
b. Since the critical test value is 1.96, we have failed to reject the null hypothesis. The rental chain store should not open a store in this area.
c. Since the critical test value is 1.96, we have failed to reject the null hypothesis. The rental chain store should not open a store in this area.
d. Since the critical test value is 1.96, we have failed to reject the null hypothesis. The rental chain store should open a store in this area.

Step-by-step explanation:

Population proportion, p = 5000/20000 = 0.25

We would set up the hypothesis test.

For the null hypothesis,

p = 0.25

For the alternative hypothesis,

p > 0.25

Considering the population proportion, probability of success, p = 0.25

q = probability of failure = 1 - p

q = 1 - 0.25 = 0.75

Considering the sample,

Sample proportion, P = x/n

Where

x = number of success = 96

n = number of samples = 300

P = 96/300 = 0.32

We would determine the test statistic which is the z score

z = (P - p)/√pq/n

z = (0.32 - 0.25)/√(0.25 × 0.75)/300 = 2.8

The calculated test statistic is 2.8 for the right tail and - 2.8 for the left tail

Since α = 0.025, the critical value is determined from the normal distribution table. The z score for the lower and upper tail of the curve is - 1.96 and 1.96

In order to reject the null hypothesis, the test statistic must be smaller than - 1.96 or greater than 1.96

Since - 2.8 < - 1.96 and 2.8 > 1.96, we would reject the null hypothesis.

Therefore, the correct option is

a. Since the critical test value is 1.96, we are going to reject the null hypothesis. The rental chain store should open a store in this area.