Chapter 22, Problem 19E

(a)

To determine

To construct: the 95 percent confidence interval for the proportion difference on basis of provided information.

Answer to Problem 19E

(-0.0649, 0.2212)

Explanation of Solution

Given:

  $\begin{array}{c}{n}_1=284\\ y_1=54\\ n_2=41\\ y_2=11\end{array}$

Formula used:

  $\begin{array}{c}SE\left({\widehat{p}}_2-{\widehat{p}}_1\right)=\sqrt{\frac{{\widehat{p}}_1{\widehat{q}}_1}{n_1}+\frac{{\widehat{p}}_2{\widehat{q}}_2}{n_2}}\\ ME=z^{\ast }\times SE\left({\widehat{p}}_2-{\widehat{p}}_1\right)\end{array}$

For the confidence interval

  $\left({\widehat{p}}_2-{\widehat{p}}_1\right)\pm ME$

Calculation:

The proportions are

  $\begin{array}{c}{\widehat{p}}_1=\frac{54}{284}=0.1901\\ {\widehat{p}}_2=\frac{11}{41}=0.2683\end{array}$

Estimating the standard deviation

  $\begin{array}{c}SE\left({\widehat{p}}_2-{\widehat{p}}_1\right)=\sqrt{\frac{{\widehat{p}}_1{\widehat{q}}_1}{n_1}+\frac{{\widehat{p}}_2{\widehat{q}}_2}{n_2}}\\ =\sqrt{\frac{0.1901\left(1-0.1901\right)}{284}+\frac{0.2683\left(1-0.2683\right)}{41}}\\ =\sqrt{\frac{0.1901\left(0.8099\right)}{284}+\frac{0.2683\left(0.7317\right)}{41}}\\ =0.0730\end{array}$

For 95% confidence level, $z^{\ast }=1.96$ , so

  $\begin{array}{c}ME=z^{\ast }\times SE\left({\widehat{p}}_2-{\widehat{p}}_1\right)\\ =1.96\times 0.0730\\ =0.1431\end{array}$

So, the 95% confidence interval is

  $\begin{array}{c}\left({\widehat{p}}_2-{\widehat{p}}_1\right)\pm ME=\left(0.2683-0.1901\right)\pm 0.1431\\ =0.0782\pm 0.1431\\ =\left(-0.0649,0.2212\right)\end{array}$

For the difference in the proportion of those who may smoke and have approving parents and those who may smoke and disapproving parents, a 95 percent confidence interval is (-0.0649, 0.2212).

(b)

To determine

To Explain: the confidence interval.

Explanation of Solution

Based on all these samples, there are 95 percent sure that the true difference is between -0.0649 and 0.2212 in the proportion of children who can smoke and have approved parents and those who can smoke and have disapproved parents. It can infer that 0 is a plausible value of the difference, since 0 lies in the interval, and there is no difference in the proportion of children who might just smoke and have parents who approve and those who might smoke and have parents who disapprove.

(c)

To determine

To Explain: the meaning of 95 percent confidence.

Explanation of Solution

Here, 95% confidence means that if it regularly take random samples of the same sizes, 95% of the intervals would represent the true difference in the proportion of children who might smoke and have parents who approve and those who may smoke and have parents who disapprove.