Chapter 22, Problem 30E

To determine

To Explain: that this provides evidence that there is a gap in the retention rates of first-year students at colleges.

Explanation of Solution

Given:

  $\begin{array}{l}{n}_{{}_{Private}}=1139\\ {\widehat{p}}_{{}_{Private}}=0.749\\ n_{Public}=505\\ {\widehat{p}}_{Public}=0.719\end{array}$

Formula used:

  $\begin{array}{c}{\widehat{p}}_{Pooled}=\frac{y_{Private}+y_{Public}}{n_{Pravite}+n_{Public}}\\ S{E}_{Pooled}\left({\widehat{p}}_{Private}-{\widehat{p}}_{Public}\right)=\sqrt{\frac{{\widehat{p}}_{Pooled}{\widehat{q}}_{Pooled}}{n_{Private}}+\frac{{\widehat{p}}_{Pooled}{\widehat{q}}_{Pooled}}{n_{Public}}}\end{array}$

Calculation:

Here, there is required to evaluate whether the retention rates of the 1^st-year students at public and private colleges are different. Let $p_{{}_{Private}}$ in private colleges represent the retention rate, and $p_{{}_{Private}}$ in public colleges represent the retention rate. If the retention rates of first-year students at private and public colleges vary, then $p_{{}_{Private}}$ should not be equal to $p_{{}_{Private}}$ . Thus, our hypothesis will be

  $\begin{array}{c}{H}_0:p_{{}_{Private}}-p_{Public}=0\\ H_1:p_{{}_{Private}}-p_{Public}\ne 0\end{array}$

  $\begin{array}{c}{y}_{Private}=1139\times 0.749=853\\ y_{Public}=505\times 0.719=363\end{array}$

Pool the sample data

  $\begin{array}{c}{\widehat{p}}_{Pooled}=\frac{y_{Private}+y_{Public}}{n_{Pravite}+n_{Public}}\\ =\frac{853+363}{1139+505}\\ =\frac{1216}{1644}\\ =0.7397\end{array}$

Estimating $SD\left(p_{Private}-p_{Public}\right)$ . Therefore,

  $\begin{array}{c}S{E}_{Pooled}\left({\widehat{p}}_{Private}-{\widehat{p}}_{Public}\right)=\sqrt{\frac{{\widehat{p}}_{Pooled}{\widehat{q}}_{Pooled}}{n_{Private}}+\frac{{\widehat{p}}_{Pooled}{\widehat{q}}_{Pooled}}{n_{Public}}}\\ =\sqrt{\frac{0.7397\left(1-0.7397\right)}{1139}+\frac{0.7397\left(1-0.7397\right)}{505}}\\ =\sqrt{\frac{0.7397\left(0.2603\right)}{1139}+\frac{0.7397\left(0.2603\right)}{505}}\\ =0.0235\end{array}$

Observed difference is

  $\begin{array}{c}{\widehat{p}}_{Private}-{\widehat{p}}_{Public}=0.749-0.719\\ =0.03\end{array}$

The z-score for the observed difference in proportions is

  $\begin{array}{c}z=\frac{\left({\widehat{p}}_{Private}-{\widehat{p}}_{Public}\right)-0}{S{E}_{Pooled}\left({\widehat{p}}_{Private}-{\widehat{p}}_{Public}\right)}\\ =\frac{0.03-0}{0.0235}\\ =1.28\end{array}$

The P-value is given as

  $\begin{array}{c}\text{P-value =2}P\left(z>1.28\right)\\ =2\left(0.0998\right)\\ =0.1996\end{array}$

To consider the null hypothesis, p-value = 0.1996 is high sufficient. It may assume that the retention rates of first year students at public and private colleges do not differ.