Chapter 21, Problem 12E

(a)

To determine

To construct a $95\%$ confidence interval.

Answer to Problem 12E

  $(0.190,0.251)$ .

Explanation of Solution

It is given in the question that the spike network commissioned a telephone poll of randomly selected U.S. men. Thus, we have,

Let us first check the conditions for the following as:

Random condition: It is satisfied as it is assumed that the sample is randomly selected.

Normal condition: It is satisfied as,

  $\begin{array}{l}np=157>10\\ n(1-p)=555>10\end{array}$

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

Thus, all the conditions are met. Therefore, now, to find the confidence interval we will use the calculator $\text{TI}-89$ . So, we have to first go to the $\text{Ints}$ menu and select $5:1-\text{PropZInt}$ . Then we have to enter the number of successes observed and the sample size. Then specify a confidence level. Then the interval will be calculated. Thus, we have,

  $\begin{array}{l}x=157\\ n=712\\ c=0.95\end{array}$

Thus, the confidence interval will be as:

  $(0.190,0.251)$

(b)

To determine

To use the confidence interval to test the appropriate hypothesis.

Answer to Problem 12E

The company should not by ads.

Explanation of Solution

It is given in the question that the spike network commissioned a telephone poll of randomly selected U.S. men. Thus, the confidence interval will be as:

  $(0.190,0.251)$

Thus, by using the confidence interval calculated in part (a), we can say that since the confidence interval extends below $25\%$ , we cannot be sure that over $25\%$ of men stay at home. Therefore, the company should not by ads.

(c)

To determine

To explain could Spike claim to the advertiser that it is possible that $25\%$ of men with young children are stay-at-home dads and what is wrong with the reasoning.

Answer to Problem 12E

Yes, he can calim.

Explanation of Solution

It is given in the question that the spike network commissioned a telephone poll of randomly selected U.S. men. Thus, the confidence interval will be as:

  $(0.190,0.251)$

Thus, by using the confidence interval calculated in part (a), we can say that since the confidence interval extends below $25\%$ , we cannot be sure that over $25\%$ of men stay at home. Thus, yes, Spike could claim to the advertiser that it is possible that $25\%$ of men with young children are stay-at-home dads as $25\%$ is the confidence interval. It may be a plausible value, but we cannot prove that the null hypothesis is true.