An insurance company checks police records on 572 accidents selected at random and notes that teenagers were at the wheel in 81 of them. Complete
parts a) through d).
a) Construct the confidence interval for the percentage of all auto accidents that involve teenage drivers.

(Round to one decimal place as needed.)
b) Explain what your interval means.
A. We are confident that the true percentage of accidents involving teenagers falls inside the confidence interval limits.
B. We are confident that a randomly sampled accident would involve a teenager a percent of the time that falls inside the confidence interval
limits.
C. We are confident that the percent of accidents involving teenagers is .

Ask by Young Warren. in the United States

Mar 28,2025

Question

An insurance company checks police records on 572 accidents selected at random and notes that teenagers were at the wheel in 81 of them. Complete
parts a) through d).
a) Construct the confidence interval for the percentage of all auto accidents that involve teenage drivers.

(Round to one decimal place as needed.)
b) Explain what your interval means.
A. We are confident that the true percentage of accidents involving teenagers falls inside the confidence interval limits.
B. We are confident that a randomly sampled accident would involve a teenager a percent of the time that falls inside the confidence interval
limits.
C. We are confident that the percent of accidents involving teenagers is .

Ask by Young Warren. in the United States

Mar 28,2025

UpStudy AI · Accepted Answer

**a) Constructing the 95% Confidence Interval**

1. Compute the sample proportion:
$$
\hat{p} = \frac{81}{572} \approx 0.1416
$$

2. Calculate the standard error (SE) using:
$$
SE = \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} = \sqrt{\frac{0.1416 \times 0.8584}{572}} \approx \sqrt{\frac{0.1215}{572}} \approx 0.0146
$$

3. Find the margin of error (ME) with the critical value for a $$95\%$$ confidence level ($$z^* = 1.96$$):
$$
ME = 1.96 \times 0.0146 \approx 0.0286
$$

4. Construct the confidence interval:
$$
\hat{p} \pm ME = 0.1416 \pm 0.0286
$$
This gives:
$$
\text{Lower limit} \approx 0.1130 \quad \text{and} \quad \text{Upper limit} \approx 0.1702
$$
In percentage form, the $$95\%$$ confidence interval is:
$$
(11.3\%, 17.0\%)
$$

**b) Interpretation of the Confidence Interval**

The correct interpretation is:

A. We are $$95\%$$ confident that the true percentage of accidents involving teenagers falls inside the confidence interval limits.
a) The 95% confidence interval for the percentage of auto accidents involving teenage drivers is (11.3%, 17.0%). b) We are 95% confident that the true percentage of accidents involving teenagers falls within the range of 11.3% to 17.0%.

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