Chapter 21, Problem 28E

(a)

To determine

To explain what are the hypotheses.

Answer to Problem 28E

  $\begin{array}{l}{H}_0:p=0.20\\ H_a:p>0.20\end{array}$

Explanation of Solution

It is given in the question that a company is willing to renew its advertising contract with a local radio station only if the station can prove that more than $20\%$ of the residents of the city have heard the ad and recognize the company’s product. Thus, the null hypothesis states that the rate will be equal to $20\%$ . And the alternative hypothesis states that the rate will be more than $20\%$ . So, we have,

  $\begin{array}{l}{H}_0:p=0.20\\ H_a:p>0.20\end{array}$

(b)

To determine

To explain why the station plans to conduct this test using a $10\%$ significance level lowered to $5\%$ .

Explanation of Solution

It is given in the question that a company is willing to renew its advertising contract with a local radio station only if the station can prove that more than $20\%$ of the residents of the city have heard the ad and recognize the company’s product. Thus, the station plans to conduct this test using a $10\%$ significance level lowered to $5\%$ because the company wants more proof that the ad is effective.

(c)

To determine

To explain what is meant by the power of this test.

Explanation of Solution

It is given in the question that a company is willing to renew its advertising contract with a local radio station only if the station can prove that more than $20\%$ of the residents of the city have heard the ad and recognize the company’s product. Thus, the power of the test in this context means that the probability of correctly deciding that more than $20\%$ have heard the ad and recognize the product when this really is the case.

(d)

To determine

To explain why and for which level of significance will the power of this test be higher.

Answer to Problem 28E

For $10\%$ level of significance the power of this test will be higher.

Explanation of Solution

It is given in the question that a company is willing to renew its advertising contract with a local radio station only if the station can prove that more than $20\%$ of the residents of the city have heard the ad and recognize the company’s product. Thus, for $10\%$ level of significance the power of this test will be higher because there is a greater chance of rejecting the null hypothesis.

(e)

To determine

To explain will that make the risk of Type II error higher or lower.

Answer to Problem 28E

The risk of Type II error will be lower.

Explanation of Solution

It is given in the question that a company is willing to renew its advertising contract with a local radio station only if the station can prove that more than $20\%$ of the residents of the city have heard the ad and recognize the company’s product. Thus, this will make the risk of Type II error lower because a larger sample size n has more power and a smaller Type II error probability.