Chapter 22, Problem 20E

(a)

To determine

To construct: a 95 percent confidence interval for the survival rate difference.

Answer to Problem 20E

(0.0039, 0.2092)

Explanation of Solution

Given:

  $\begin{array}{l}{n}_1=89\\ y_1=26\\ n_2=361\\ y_2=67\end{array}$

Formula used:

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

For the confidence interval

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

Calculation:

Observed proportions are

  $\begin{array}{c}{\widehat{p}}_1=\frac{26}{89}=0.2921\\ {\widehat{p}}_2=\frac{67}{361}=0.1856\end{array}$

Estimating the standard deviation

  $\begin{array}{c}SE\left({\widehat{p}}_1-{\widehat{p}}_2\right)=\sqrt{\frac{{\widehat{p}}_1{\widehat{q}}_1}{n_1}+\frac{{\widehat{p}}_2{\widehat{q}}_2}{n_2}}\\ =\sqrt{\frac{0.2921\left(1-0.2921\right)}{89}+\frac{0.1856\left(1-0.1856\right)}{361}}\\ =\sqrt{\frac{0.2921\left(0.7079\right)}{89}+\frac{0.1856\left(0.8144\right)}{361}}\\ =0.0524\end{array}$

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

  $\begin{array}{c}ME=z^{\ast }\times SE\left({\widehat{p}}_1-{\widehat{p}}_2\right)\\ =1.96\times 0.0524\\ =0.1026\end{array}$

Therefore, the 95% confidence interval is

  $\begin{array}{c}\left({\widehat{p}}_2-{\widehat{p}}_1\right)\pm ME=\left(0.2921-0.1856\right)\pm 0.1026\\ =0.1065\pm 0.1026\\ =\left(0.0039,0.2092\right)\end{array}$

A 95 % confidence interval is (0.0039, 0.2092) for the difference in the mortality rates of patients with mild or severe depression and no depression.

(b)

To determine

To Explain: the confidence interval.

Explanation of Solution

Based on this samples, we are 95 percent sure that the real difference is between 00039 and 02092 in the mortality rates of patients with minor or major depression and no depression. This period shows that 04% to 21% of depressed patients are more likely to suffer than non-depressed patients.

(c)

To determine

To Explain: the meaning of 95 percent confidence.

Explanation of Solution

Here, 95 percent confidence' means that if random samples with the similar sizes are frequently taken, 95 percent with the intervals would catch a true difference in the mortality rates of patients with mild or severe depression and no depression.