Questions: An insurance company checks police records on 553 accidents selected at random and notes that teenagers were at the wheel in 96 of them. Complete parts a) through d). a) Construct the 95% confidence interval for the percentage of all auto accidents that involve teenage drivers. 95% CI=(%, %) (Round to one decimal place as needed.)

Solution Steps

The sample proportion of accidents involving teenage drivers is calculated as follows:

\[ \hat{p} = \frac{x}{n} = \frac{96}{553} \approx 0.1736 \]

To construct the \(95\%\) confidence interval for the population proportion, we use the formula:

\[ \hat{p} \pm z \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \]

Where:

• \(\hat{p} \approx 0.174\)
• \(z \approx 1.96\) (for \(95\%\) confidence level)
• \(n = 553\)

Calculating the margin of error:

\[ \text{Margin of Error} = z \sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} = 1.96 \cdot \sqrt{\frac{0.174(1 - 0.174)}{553}} \approx 0.031 \]

Thus, the confidence interval is:

\[ (0.174 - 0.031, 0.174 + 0.031) = (0.142, 0.205) \]

Converting the confidence interval to percentage:

\[ (0.142 \times 100, 0.205 \times 100) = (14.2\%, 20.5\%) \]

Final Answer