Chapter 22, Problem 13E

(a)

To determine

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

Answer to Problem 13E

Yes

Explanation of Solution

It is told that samples are split into two classes on a random basis. The total size of the sample is less than 10% of all children. It is fair to assume that both groups are independent, since the samples were split at random. The number of successes among vaccinated children is 333 and the number of failures is 2455-333=2122 and the number of successes among non-vaccinated children is 499 and the number of failures is 2452-499=1953. 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 ear rates of infection.

Answer to Problem 13E

(0.0470, 0.0888)

Explanation of Solution

Given:

  $n_1=2455$

  $n_2=2452$

Formula used:

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

Calculation:

the observed proportions are

  $\begin{array}{l}{\widehat{p}}_1=\frac{333}{2455}=0.1356\\ {\widehat{p}}_2=\frac{499}{2452}=0.2035\end{array}$

Estimating the standard deviation

  $\begin{array}{c}E\left({\widehat{p}}_2-{\widehat{p}}_1\right)=\sqrt{\frac{p_1{q}_1}{n_1}+\frac{p_2{q}_2}{n_2}}\\ =\sqrt{\frac{0.1356\left(1-0.1356\right)}{2455}+\frac{0.2035\left(1-0.2035\right)}{2452}}\\ =\sqrt{\frac{0.1356\left(0.8644\right)}{2455}+\frac{0.2035\left(0.7965\right)}{2452}}\\ =0.0107\end{array}$

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

  $\begin{array}{c}ME=z^{\ast }\times SE\left({\widehat{p}}_2-{\widehat{p}}_1\right)\\ =1.96\times 0.0107\\ =0.0209\end{array}$

Therefore, the 95% confidence interval is

  $\begin{array}{c}\left({\widehat{p}}_2-{\widehat{p}}_1\right)\pm ME=\left(0.2035-0.1356\right)\pm 0.0209\\ =0.0679\pm 0.0209\\ =\left(0.0470,0.0888\right)\end{array}$

A 95 % confidence interval is (0.0470, 0.0888) for the difference in the proportion of unvaccinated and vaccinated children who have ear infections.

(c)

To determine

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

Explanation of Solution

There are 95 percent sure, based on these samples, that the real difference in the proportion of ear infections among unvaccinated and vaccinated children is between 4.7 percent and 8.88 percent. Since 0 is not in interval and both values are positive, it can infer that the percentage of ear infections among unvaccinated children is 4.7 to 8.88 percent higher than that of vaccinated children. it may assume, then, that the vaccine is effective.