Chapter 10.1, Problem 10E

(a)

To determine

To construct and interpret a $95\%$ confidence interval for the difference in the true proportion of U.S. men and U.S. women who can identify Egypt on a map.

Answer to Problem 10E

The confidence interval is $(-0.0191,\text{0}\text{.1017)}$ .

Explanation of Solution

Given:

  $\begin{array}{l}{x}_1=287\\ x_2=233\\ n_1=522\\ n_2=520\\ c=95\%=0.95\end{array}$

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 $522$ U.S. men are less than $10\%$ of all U.S. men and the same is for the U.S. women also.

Normal: It is satisfied because there are $287$ successes and $522-287=235$ failures in the first sample and there are $233$ successes and $520-233=287$ 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,

  $\begin{array}{l}{\widehat{p}}_1=\frac{x_1}{n_1}\\ =\frac{287}{522}\\ =0.5498\\ {\widehat{p}}_2=\frac{x_2}{n_2}\\ =\frac{233}{520}\\ =0.4481\end{array}$

Now, for the confidence interval $1-\alpha =0.95$ , we have to find out the value of $z_{\alpha /2}=z_{0.025}$ using the table. Thus the z score will be:

  $z_{\alpha /2}=1.96$

Thus, the confidence interval will be:

  $\begin{array}{l}({\widehat{p}}_1-{\widehat{p}}_2)-z_{\alpha /2}\times \sqrt{\frac{{\widehat{p}}_1(1-{\widehat{p}}_1)}{n_1}+\frac{{\widehat{p}}_2(1-{\widehat{p}}_2)}{n_2}}\\ =(0.5498-0.4481)-1.96\times \sqrt{\frac{0.5498(1-0.5498)}{522}+\frac{0.4481(1-0.4481)}{520}}\\ \approx -0.0191\\ ({\widehat{p}}_1-{\widehat{p}}_2)+z_{\alpha /2}\times \sqrt{\frac{{\widehat{p}}_1(1-{\widehat{p}}_1)}{n_1}+\frac{{\widehat{p}}_2(1-{\widehat{p}}_2)}{n_2}}\\ =(0.5498-0.4481)+1.96\times \sqrt{\frac{0.5498(1-0.5498)}{522}+\frac{0.4481(1-0.4481)}{520}}\\ \approx 0.1017\end{array}$

Thus, we conclude that, we are $95\%$ confident that the true proportion of U.S. men who can identify Egypt on a map is between lower $0.0191\text{ and 0}\text{.1017}$ higher than the true proportion of U.S. women who can identify Egypt on a map.

(b)

To determine

To explain based on your interval, is there a convincing evidence of a difference in the true proportions of U.S. men and women who can identify Egypt on a map or not.

Answer to Problem 10E

Yes, there is.

Explanation of Solution

The confidence interval in part (a) is calculated as:

  $(-0.0191,\text{0}\text{.1017)}$

Thus, this confidence interval does not contain zero, then it is very unlikely that the population proportions are equal and thus there is a convincing evidence of a difference between the population proportions. Thus, confidence interval calculated in part (a) gives the convincing evidence.