Chapter 20, Problem 20.1AYK

To determine

To give a $95\%$ confidence interval for the difference between the proportions of smokers who still abstain from smoking six months after they stopped when quitting either abruptly or gradually.

Answer to Problem 20.1AYK

  $(0.0071,0.1224)$ .

Explanation of Solution

In the question, it is given that verified abstinence was achieved at that point by $\text{78 of 355}$ subjects in the abrupt-cessation group and $\text{53 of the 342}$ subjects in the gradual-cessation group. Thus, we know that the subjects were chosen randomly for the study and also the conditions for large-sample is satisfied because we have,

  $\begin{array}{l}{x}_1\times p_1>10\\ x_2\times p_2>10\end{array}$

Thus, the proportions are as follows:

  $\begin{array}{l}{\widehat{p}}_1=\frac{x_1}{n_1}\\ =\frac{78}{355}\\ =0.2197\\ {\widehat{p}}_2=\frac{x_2}{n_2}\\ =\frac{53}{342}\\ =0.1550\end{array}$

Thus, the $95\%$ confidence interval for the difference between the proportions of smokers can be calculated as:

  $\begin{array}{l}SE=\sqrt{\frac{{\widehat{p}}_1(1-{\widehat{p}}_1)}{n_1+2}+\frac{{\widehat{p}}_2(1-{\widehat{p}}_2)}{n_2+2}}\\ =\sqrt{\frac{0.2197(1-0.2197)}{355}+\frac{0.1550(1-0.1550)}{342}}\\ =0.0294\\ z_{\alpha /2}=z_{0.025}=1.96\\ \text{Confidence interval}=(0.2197-0.1550)\pm 1.96\times 0.0294\\ =0.0647\pm 0.0577\\ =(0.0071,0.1224)\end{array}$

So, we are $95\%$ confident that the difference between the proportions of smokers who still abstain from smoking six months after they stopped when quitting either abruptly or gradually is between $(0.0071,0.1224)$ .