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Question

# An insurance company checks police records on 582 accidents selected at random and notes that teenagers were at the wheel in 91 of them. a) Create a 95% confidence interval for the percentage of all auto accidents that involve teenage drivers. b) Explain what your interval means. c) Explain what “95% confidence” means. d) A politician urging tighter restrictions on drivers’ licenses issued to teens says, “In one of every five auto accidents, a teenager is behind the wheel.” Does your confidence interval support or contradict this statement? Explain.

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### a)

We are given that out of $582$ accidents selected at random, we have noted that $91$ of these accidents have teenagers at the wheel. Since we are estimating for the true percentage of all auto accidents that involve teenage drivers, we know that the sampling proportion that we have is the total proportion from the sample that had teenager drivers which is $\frac{91}{582} = 0.15636$.

To construct the $95\%$ confidence interval of the given experiment, we will use the confidence interval formula:

$\bigg( \hat{p} \pm z_{\frac{\alpha}{2}}\sqrt{\dfrac{\hat{p}\hat{q}}{n}}\bigg)$

where $\hat{p}$ is the sample proportion computed from the sample, $z_{\frac{\alpha}{2}}$ is the $z$ critical value given $\alpha$ in a $(1-\alpha)\% = 95\%$ confindence interval, and $n$ is the sample size taken for the experiment.

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