Chapter 22, Problem 14E

(a)

To determine

To explain: whether or not the conditions for inference are fulfilled.

Answer to Problem 14E

Yes

Explanation of Solution

It is told that samples are separated into 2 groups at random. The overall sample size of all women being treated for anorexia is less than 10 percent. It's fair to assume that both groups are independent since the samples were split at random. The number of successes is 35 among the 49 women who received Prozac and the number of failures is 49-35=14 and among the 44 women who received placebo, the number of successes is 32 and the number of failures is 44-32=12. These numbers, for each group, are at minimum 10.

The assumptions and conditions required for inference are met.

(b)

To determine

To find: a confidence interval of 95 percent for the difference in outcomes.

Answer to Problem 14E

(-0.1695, 0.1955)

Explanation of Solution

Given:

  $n_1=49$

  $n_2=44$

Formula used:

  $SE\left({\widehat{p}}_2-{\widehat{p}}_1\right)=\sqrt{\frac{p_1{q}_1}{n_1}+\frac{p_2{q}_2}{n_2}}$

  $ME=z^{\ast }\times SE\left({\widehat{p}}_2-{\widehat{p}}_1\right)$

For the confidence interval

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

Calculation:

Observed proportions are

  $\begin{array}{c}{\widehat{p}}_1=\frac{32}{49}=0.7143\\ {\widehat{p}}_2=\frac{32}{44}=0.7273\end{array}$

Estimating the standard deviation

  $\begin{array}{c}SE\left({\widehat{p}}_2-{\widehat{p}}_1\right)=\sqrt{\frac{p_1{q}_1}{n_1}+\frac{p_2{q}_2}{n_2}}\\ =\sqrt{\frac{0.7143\left(1-0.7143\right)}{49}+\frac{0.7273\left(1-0.7273\right)}{44}}\\ =\sqrt{\frac{0.7143\left(0.2857\right)}{49}+\frac{0.7273\left(0.2727\right)}{44}}\\ =0.0931\end{array}$

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

  $\begin{array}{c}ME=z^{\ast }\times SE\left({\widehat{p}}_2-{\widehat{p}}_1\right)\\ =1.96\times 0.0931\\ =0.1825\end{array}$

Therefore, the 95% confidence intervals is

  $\begin{array}{c}\left({\widehat{p}}_2-{\widehat{p}}_1\right)\pm ME=\left(0.7273-0.7143\right)\pm 0.1825\\ =0.0130\pm 0.1825\\ =\left(-0.1695,0.1955\right)\end{array}$

A 95 percent confidence interval is (-0.1695, 0.1955) for the difference in proportions of healthy women who got a placebo and got Prozac a year later.

(c)

To determine

To explain: whether the Prozac uses a confidence interval to be successful or not.

Explanation of Solution

Based on these samples, there are 95 percent sure that the real difference between -17 percent and 20 percent is the proportion of healthy women who got a placebo a year after and who got Prozac. Because 0 is in the interval, 0 is the reasonable value of the difference, it may conclude. Therefore, it may conclude that Prozac is not successful.