Chapter 9.4, Problem 17E

To determine

To Construct:

Construct and interpret a 90% confidence interval for the true differences between the percentages of Northerners and Southerners who attend church on a weekly basis.

Answer to Problem 17E

Solution:

The true differences between the percentages of Northerners and Southerners who attend church on a weekly basis is between $-57.6\%$ and $-27.5\%$ with 90% confidence. Thus with 90 % confidence a large percentage of Southerners than Northerners attend church on a weekly basis.

Explanation of Solution

Formula Used:

When the samples taken are independent, simple random samples, the conditions for a binomial distribution are met for both samples, and the sample sizes are large enough to ensure that $n_1{\widehat{p}}_1\ge 5,$$n_1(1-{\widehat{p}}_1)\ge 5,$$n_2{\widehat{p}}_2\ge 5,$$n_2(1-{\widehat{p}}_2)\ge 5$, the margin of error of a confidence interval for the differences between two population proportions is given by

$E=z_{\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\sqrt{\frac{{\widehat{p}}_1(1-{\widehat{p}}_1)}{n_1}+\frac{{\widehat{p}}_2(1-{\widehat{p}}_2)}{n_2}}$

Where $z_{\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ is the critical value for the level of significance, $c=1-\alpha$, such that the area under the standard normal distribution to the right of $z_{\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}$ is equal to $\frac{\alpha }{2}$, ${\widehat{p}}_{1}and{\widehat{p}}_2$ are two sample proportions, and $n_{1}and{n}_2$ are two sample sizes.

The confidence interval for the difference between two population proportions is given by

$\left({\widehat{p}}_1-{\widehat{p}}_2\right)-E<p_1-p_2<\left({\widehat{p}}_1-{\widehat{p}}_2\right)+E$ or $\left(\left({\widehat{p}}_1-{\widehat{p}}_2\right)-E,\left({\widehat{p}}_1-{\widehat{p}}_2\right)+E\right)$

Where ${\widehat{p}}_{1}and{\widehat{p}}_2$ are two sample proportions, and $n_{1}and{n}_2$ are two sample sizes, $\left({\widehat{p}}_1-{\widehat{p}}_2\right)$ is the point estimate for the difference between population proportions, E is the margin of error.

Calculation:

Let population 1 be the sample of Northerners and population 2 be the samples of Southerners. The sample size for the first and second sample is calculated as follows:

$\begin{array}{l}{n}_1=34+12=46\\ n_2=35+16=51\end{array}$

At 90% level of confidence so $z_{\alpha /2}=z_{0.10/2}=1.645$

Using these sample sizes to calculate the sample proportions

$\begin{array}{rll}{\widehat{p}}_1& =& \frac{x_1}{n_1}\\ & =& \frac{12}{46}\\ & =& 0.26086\end{array}$$$

$\begin{array}{rll}{\widehat{p}}_2& =& \frac{x_2}{n_2}\\ & =& \frac{35}{51}\\ & =& 0.68627\end{array}$

Subtracting these two sample proportions produces the point estimate.

$\begin{array}{rll}\left({\widehat{p}}_1-{\widehat{p}}_2\right)& =& 0.26086-0.68627\\ & =& -0.42541\end{array}$

The margin of error is calculated as follows-:

$\begin{array}{rll}E& =& z_{\raisebox{1ex}{$\alpha $}\!\left/ \!\raisebox{-1ex}{$2$}\right.}\sqrt{\frac{{\widehat{p}}_1(1-{\widehat{p}}_1)}{n_1}+\frac{{\widehat{p}}_2(1-{\widehat{p}}_2)}{n_2}}\\ & =& 1.645\sqrt{\frac{0.26086(1-0.26086)}{46}+\frac{0.68627(1-0.68627)}{51}}\\ & =& 0.15087\end{array}$

Subtracting the margin of error from the point estimate and then adding the margin of error to the point estimate gives the following endpoints of the confidence interval

Lower end point:

$\begin{array}{rll}\left({\widehat{p}}_1-{\widehat{p}}_2\right)-E& =& -0.42541-0.15087\\ & =& -0.57628\end{array}$

Upper end point:

$\begin{array}{rll}\left({\widehat{p}}_1-{\widehat{p}}_2\right)+E& =& -0.42541+0.15087\\ & =& -0.27454\end{array}$

Thus, the 90% confidence interval for the difference between the two population ranges from -0.57628 to -0.27454. The confidence interval can be written mathematically using either inequality symbols or interval notation, as follows.

$-0.57628<p_1-p_2<-0.27454$ or $(-0.57628,-0.27454)$

Convert into percentage:

$(-0.57628\times 100,-0.27454\times 100)$

$(-57.6\%,-27.5\%)$

Interpretation:

The true differences between the percentages of Northerners and Southerners who attend church on a weekly basis is between $-57.6\%$ and $-27.5\%$ with 90% confidence. Thus with 90 % confidence a large percentage of Southerners than Northerners attend church on a weekly basis.