Chapter 22, Problem 29E

To determine

To explain: that Time magazines are justified by the numbers given.

Explanation of Solution

Given:

  $\begin{array}{c}{n}_Y=129\\ y_Y=80\\ n_o=184\\ y_O=98\end{array}$

Formula used:

  $\begin{array}{c}{\widehat{p}}_{Pooled}=\frac{y_Y+y_O}{n_Y+n_O}\\ S{E}_{Pooled}\left({\widehat{p}}_Y-{\widehat{p}}_{{}_O}\right)=\sqrt{\frac{{\widehat{p}}_{Pooled}{\widehat{q}}_{Pooled}}{n_Y}+\frac{{\widehat{p}}_{Pooled}{\widehat{q}}_{Pooled}}{n_O}}\end{array}$

Calculation:

Here, there is required to check whether young men are more comfortable to talk about their issues than adult men. Let $p_Y$ represent the proportion of men between the ages of 18 and 24, and $p_O$ represent the proportion of men between the ages of 25 and 34 who are comfortable to talk about their issues. If young men are more comfortable to talk about their issues than older men, then $p_Y$ should be larger than $p_O$

Therefore, the hypotheses would be

  $\begin{array}{c}{H}_0:p_Y-p_O=0\\ H_1:p_Y-p_O>0\end{array}$

  $\begin{array}{c}{\widehat{p}}_Y=\frac{80}{129}=0.62,\\ {\widehat{p}}_O=\frac{98}{184}=0.53\end{array}$

Therefore, Pool the sample data

  $\begin{array}{c}{\widehat{p}}_{Pooled}=\frac{y_Y+y_O}{n_Y+n_O}\\ =\frac{80+98}{129+184}\\ =\frac{178}{313}\\ =0.5687\end{array}$

Using the pooled SE to estimate $SD\left({\widehat{p}}_Y-{\widehat{p}}_{{}_O}\right)$ . Thus,

  $\begin{array}{c}S{E}_{Pooled}\left({\widehat{p}}_Y-{\widehat{p}}_{{}_O}\right)=\sqrt{\frac{{\widehat{p}}_{Pooled}{\widehat{q}}_{Pooled}}{n_Y}+\frac{{\widehat{p}}_{Pooled}{\widehat{q}}_{Pooled}}{n_O}}\\ =\sqrt{\frac{0.5687\left(1-0.5687\right)}{129}+\frac{0.5687\left(1-0.5687\right)}{184}}\\ =\sqrt{\frac{0.5687\left(0.4313\right)}{129}+\frac{0.5687\left(0.4313\right)}{184}}\\ =0.0569\end{array}$

The observed difference is

  $\begin{array}{c}{\widehat{p}}_Y-{\widehat{p}}_{{}_O}=0.62-0.53\\ =0.09\end{array}$

The z-score is

  $\begin{array}{c}z=\frac{\left({\widehat{p}}_Y-{\widehat{p}}_{{}_O}\right)-0}{S{E}_{Pooled}\left({\widehat{p}}_Y-{\widehat{p}}_{{}_O}\right)}\\ =\frac{0.09-0}{0.0569}\\ =1.54\end{array}$

The P-value is

  $\begin{array}{c}\text{P-value =}P\left(z>1.54\right)\\ =0.0619\end{array}$

It may reject the null hypothesis. P-value = 0.0619 is small. There isn't really good evidence, but it can still say that young men are more comfortable to talk about their issues than adult people.