Chapter 21, Problem 30E

(a)

To determine

To explain should the company renew the contract.

Answer to Problem 30E

The company should not renew the contract.

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,

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

Thus, let us check the appropriate conditions and assumptions for inference to proceed further to test hypothesis as:

Random condition: It is satisfied as we assumed that the sample was randomly selected from the population.

Independent condition: It is satisfied as we can assume that the sample is independent.

  $10\%$ condition: It is satisfied as the sample is less than $10\%$ of all population.

Success/failure condition: It is satisfied because both are greater than ten as,

  $\begin{array}{l}np=133>10\\ n(1-p)=467>10\end{array}$

Thus, all the conditions are satisfied. Thus, for this first we have to select $5:1-\text{PropZInt}$ from the STAT TESTS menu. Then, enter the observed count and sample size for the sample. Then we will indicate what kind of test we want: one-tail lower tail, two-tail, or one-tail upper tail. Then, we will specify whether to calculate the result or draw the result. Thus, we have,

  $\begin{array}{l}x=133\\ n=600\\ c=0.95=95\%\end{array}$

So, by putting the values above by using the calculator we will get the result as,

The value of z-statistics and the P-value will be then,

  $\begin{array}{l}z=1.33\\ P-\text{value}=0.092\end{array}$

And as we know if the P-value is smaller than the significance level, then reject the null hypothesis, so, we have,

  $P>0.05\Rightarrow \text{ Fail to Reject }{H}_0$

Thus, we conclude that we are unable to prove that over $20\%$ of residents remember the ad. So, the company should not renew the contract.

(b)

To determine

To explain what your P-value means in this context.

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.

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

The value of z-statistics and the P-value will be then,

  $\begin{array}{l}z=1.33\\ P-\text{value}=0.092\end{array}$

Thus, the P-value in this context means that there is a $9.23\%$ chance of having $133$ or more of $600$ people in a random sample remember the as if in fact, $20\%$ of people do.