Question

The Fox TV network is considering replacing one of its prime-time crime investigation shows with a new family-oriented comedy show. Before a final decision is made, network executives designed an experiment to estimate the proportion of their viewers who would prefer the comedy show over the crime investigation show. A random sample of 400 viewers was selected and asked to watch the new comedy show and the crime investigation show. After viewing the shows, 250 indicated they would watch the new comedy show and suggested it replace the crime investigation show.

Required:
a. Estimate the value of the population proportion.
b. Develop a 99 percent confidence interval for the population proportion.
c. Interpret your findings.

Answer 1

a) p = 0.625

b) (0.563, 0.687)

c) This means that we are 99% sure that the true proportion of all viewers who would prefer the comedy show over the crime investigation show is between (0.563, 0.687).

Explanation:

a. Estimate the value of the population proportion.

Sample: 250 viewers would prefer the comedy show over the crime investigation show, out of 400 viewers sampled. So

`p = (250)/(400) = 0.625`

b. Develop a 99 percent confidence interval for the population proportion.

In a sample with a number n of people surveyed with a probability of a success of
`\pi`, and a confidence level of
`1-\alpha`, we have the following confidence interval of proportions.

`\pi \pm z\sqrt{(\pi(1-\pi))/(n)}`

In which

z is the zscore that has a pvalue of
`1 - (\alpha)/(2)`.

For this problem, we have that:

`n = 400, \pi = 0.625`

99% confidence level

So
`\alpha = 0.01`, z is the value of Z that has a pvalue of
`1 - (0.01)/(2) = 0.995`, so
`Z = 2.575`.

The lower limit of this interval is:

`\pi - z\sqrt{(\pi(1-\pi))/(n)} = 0.625 - 2.575\sqrt{(0.625*0.375)/(400)} = 0.563`

The upper limit of this interval is:

`\pi + z\sqrt{(\pi(1-\pi))/(n)} = 0.625 + 2.575\sqrt{(0.625*0.375)/(400)} = 0.687`

c. Interpret your findings.

This means that we are 99% sure that the true proportion of all viewers who would prefer the comedy show over the crime investigation show is between (0.563, 0.687).