Chapter 10.1, Problem 12E

To determine

To construct and interpret a $99\%$ confidence interval for the difference in the true proportion of smokers like these who would abstain when using bupropion and a nicotine patch and the proportion who would abstain when using only a patch.

Answer to Problem 12E

The confidence interval is $(-0.2908,-0.0916)$ .

Explanation of Solution

Given:

  $\begin{array}{l}{x}_1=40\\ x_2=87\\ n_1=244\\ n_2=245\\ c=99\%=0.99\end{array}$

Conditions to be satisfied:

There are three conditions to be satisfied:

Random: It is satisfied because the samples are independent random samples.

Independent: It is satisfied because the $244$ smokers are less than $10\%$ of all smokers and $245$ smokers are less than $10\%$ of all smokers.

Normal: It is satisfied because there are $40$ successes and $244-40=204$ failures in the first sample and there are

  $87$ successes and $245-87=158$ failures in the second sample, which are all at least ten.

Thus, all the conditions are satisfied.

Calculation:

The sample proportion is the number of successes divided by the sample size. Then, we have,

  $\begin{array}{l}{\widehat{p}}_1=\frac{x_1}{n_1}\\ =\frac{40}{244}\\ =0.1639\\ {\widehat{p}}_2=\frac{x_2}{n_2}\\ =\frac{87}{245}\\ =0.3551\end{array}$

Now, for the confidence interval $1-\alpha =0.99$ , we have to find out the value of $z_{\alpha /2}=z_{0.005}$ using the table. Thus the z score will be:

  $z_{\alpha /2}=2.575$

Thus, the confidence interval will be:

  $\begin{array}{l}({\widehat{p}}_1-{\widehat{p}}_2)-z_{\alpha /2}\times \sqrt{\frac{{\widehat{p}}_1(1-{\widehat{p}}_1)}{n_1}+\frac{{\widehat{p}}_2(1-{\widehat{p}}_2)}{n_2}}\\ =(0.1639-0.3551)-2.575\times \sqrt{\frac{0.1639(1-0.1639)}{244}+\frac{0.3551(1-0.3551)}{245}}\\ \approx -0.2908\\ ({\widehat{p}}_1-{\widehat{p}}_2)+z_{\alpha /2}\times \sqrt{\frac{{\widehat{p}}_1(1-{\widehat{p}}_1)}{n_1}+\frac{{\widehat{p}}_2(1-{\widehat{p}}_2)}{n_2}}\\ =(0.1639-0.3551)+2.575\times \sqrt{\frac{0.1639(1-0.1639)}{244}+\frac{0.3551(1-0.3551)}{245}}\\ \approx -0.0916\end{array}$

Thus, we conclude that, we are $99\%$ confident that the true proportion of smokers who use nicotine patch and abstained from smoking after a year is between $0.2908\text{ and 0}\text{.0916}$ lower than the true proportion of smokers who use the patch and also used the drugs and abstained from smoking after a year.