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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 582582 accidents selected at random, we have noted that 9191 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 91582=0.15636\frac{91}{582} = 0.15636.

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

(p^±zα2p^q^n)\bigg( \hat{p} \pm z_{\frac{\alpha}{2}}\sqrt{\dfrac{\hat{p}\hat{q}}{n}}\bigg)

where p^\hat{p} is the sample proportion computed from the sample, zα2z_{\frac{\alpha}{2}} is the zz critical value given α\alpha in a (1α)%=95%(1-\alpha)\% = 95\% confindence interval, and nn is the sample size taken for the experiment.

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