Chapter 9.2, Problem 53E

(a)

To determine

To construct: and interpret a 95% confidence interval for the true proportion of all teens in the state that passed their driving test on the first attempt.

Answer to Problem 53E

  $0.6068<p<0.7692$

There are 95% confident that the true proportion of all teens in the state that passed their driving test on the first attempt is between 0.6068 and 0.7692.

Explanation of Solution

Given:

  $\begin{array}{l}n=125\\ p=60\%=0.60\\ x=86\\ c=95\%\end{array}$

Formula used:

  $\begin{array}{c}\widehat{p}=\frac{x}{n}\\ E=z_{\alpha /2}\cdot \sqrt{\frac{\widehat{p}\left(1-\widehat{p}\right)}{n}}\end{array}$

Calculation:

The sample proportion is

  $\widehat{p}=\frac{x}{n}=\frac{86}{125}=0.688$

For confidence level $1-\alpha =0.95$ , determine $z_{\alpha /2}=z_{0.025}$ using table II (look up 0.025 in the table, the z-score is then they found z-core with opposite sign):

  $z_{\alpha /2}=1.96$

The margin of error is

  $\begin{array}{c}E=z_{\alpha /2}\cdot \sqrt{\frac{\widehat{p}\left(1-\widehat{p}\right)}{n}}\\ =1.96\times \sqrt{\frac{0.688\left(1-0.688\right)}{125}}\\ =0.0812\end{array}$

The confidence interval is

  $\begin{array}{c}\widehat{p}-E<p<\widehat{p}+E=0.688-0.0812<p<0.688+0.0812\\ =0.6068<p<0.7692\end{array}$

There are 95% confident that the true proportion of all teens in the state that passed their driving test on the first attempt is between 0.6068 and 0.7692.

(b)

To determine

To Explain: the interval in part (a) provides more information than the test exercise 51.

Answer to Problem 53E

There is sufficient proof to reject the DMV’s claim.

Explanation of Solution

Given:

Result from exercise part (a):

  $0.6068<p<0.7692$

The confidence interval does not having 60% (or 0.60) which is what the DMV claimed. Therefore there is sufficient proof to reject the DMV’s claim.