Chapter 18, Problem 1P

1. Insurance companies charge young drivers more for automobile insurance because they tend to have more accidents than older drivers. To make this point an insurance representative first determines that only 15% of licensed drivers are age 20 or younger.

Because this age group makes up only 15% of the drivers, it is reasonable to predict that they should be involved in only 15% of the accidents. In a random sample of 80 accident reports, however, the represen­ tative finds 26 accidents that involved drivers who were 20 or younger. Is this sample sufficient to show that younger drivers have significantly more accidents than would be expected from the percentage of young drivers? Use a two-tailed test with a = .05.

To determine

If the sample sufficient to show that younger drivers have significantly more accidents than would be expected from the percentage of young drivers using a two-tailed test with $\alpha =0.05$.

Answer to Problem 1P

Solution:

Since np and nq = 12 >5 so using the Binomial test and Z scores is valid with $\mu =n*p,\sigma =\sqrt{n*p*q},q=1-p$

$\begin{array}{l}{H}_0:p_{}=0.15,H_1:p_{}\ne 0.15,n=80\\ {\widehat{p}}_{}=26/80=.325\\ Z_{calculated}=\frac{\widehat{p}-p_{}}{\sqrt{p_{}*q_{}/n}}=4.3835\\ \end{array}$

$Z_{tabulated,0.05,2tailed}=(-1.96,+1.96)$ Hence $H_0$ of p=0.15 is rejected at alpha=0.05 hence the sample is sufficient to show that younger drivers have significantly more accidents than would be expected from the percentage of young drivers.

Explanation of Solution

Given:

Insurance companies charge young drivers more for automobile insurance because they tend to have more accidents than older drivers. In a random sample of 80 accident reports, however, the representative finds 26 accidents that involved drivers who were 20 or younger.

Calculations:

The null hypothesis states that proportion of accidents in population is same as proportion of young drivers(15%). $H_0:p=0.15,H_1:p\ne 0.15$ .To test this we construct the Z score as np = nq = 12 > 10. $Z_{calculated}=\frac{\widehat{p}-p_{}}{\sqrt{p*q/n}}=4.3835$, $Z_{tabulated}=1.96$ and which is less than $Z_{calculated}=4.3835$ hence Null is rejected.

Conclusion:

$H_0$ Of p =0.15 is rejected which implies there's evidence from the sample that younger drivers have significantly more accidents than would be expected from the percentage of young drivers.

Justification:

The test concludes that the null hypotheses about the population proportion of 15% of licensed young drivers who are involved in accidents are actually significantly different from 15%. The sample has actually 32.5% of young drivers who commits accident. The sample has enough evidence to refute the hypothesis of 15%.