Question

Teen drivers Refer to Exercise 51.

a. Construct and interpret a 95% confidence interval for the true proportion p of all teens in the state who passed their driving test on the first attempt. Assume that the conditions for inference are met.

b. Explain why the interval in part (a) provides more information than the test in Exercise 51.

Short answer

Part a) $0.6068<p<0.7692$

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 $0.7692$

Part b) There is sufficient proof to reject the DMV's claim.

Step-by-step solution

Part a) Step 1: given information

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

Part a) Step 2: Formula used

$$\begin{gathered} \hat{p}=\frac{x}{n} \\ E=z_{\frac{a}{2}}·\sqrt{\frac{\hat{p}(1-\hat{p}}{n}} \end{gathered}$$

Part a) Step 3: The objective is to construct and interpret a  confidence interval for the true proportion of all teens in the state that passed their driving test on the first attempt.

The sample proportion is

$$\begin{gathered} \hat{p}=\frac{x}{n} \\ =\frac{86}{125}=0.688 \end{gathered}$$

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

$z_{\alpha /2}=1.96$

The margin of error is:

$$\begin{gathered} E=z_{a/2}·\sqrt{\frac{\hat{p}(1-\hat{p})}{n}} \\ =1.96\times \sqrt{\frac{0.688(1-0.688)}{125}} \\ =0.0812 \end{gathered}$$

The confidence interval is as follows:

$$\begin{gathered} \hat{p}-E<p<\hat{p}+E=0.688-0.0812<p<0.688+0.0812 \\ =0.6068<p<0.7692 \end{gathered}$$

The true proportion of all teens in the state who passed their driving test on the first try is estimated to be between $0.6068$ and $0.7692$

Part b) Step 1: Given information

The result from exercise part (a):

$0.6068<p<0.7692$

Part b) Step 2: The objective is to explain the interval in part (a) provides more information than the test exercise 51

The confidence interval is not $60\%$ (or $0.60$), as claimed by the DMV. As a result, there is sufficient evidence to reject the DMV's claim.