Question

Teenagers (age 15 to 20 ) make up \(7 \%\) of the driving population. The article "More States Demand Teens Pass Rigorous Driving Tests" (San Luis Obispo Tribune, January 27,2000 ) described a study of auto accidents conducted by the Insurance Institute for Highway Safety. The Institute found that \(14 \%\) of the accidents studied involved teenage drivers. Suppose that this percentage was based on examining records from 500 randomly selected accidents. Does the study provide convincing evidence that the proportion of accidents involving teenage drivers differs from \(.07\), the proportion of teens in the driving population?

Short answer

The answer depends on the computed p-value. If it's smaller than 0.05, it provides convincing evidence that the proportion of accidents involving teenage drivers significantly differs from the proportion of teenagers in the driving population (0.07). If it's larger, then it doesn't provide convincing evidence.

Step-by-step solution

Formulate the Hypotheses

The null hypothesis (\(H_0\)) is that the proportion of accidents involving teenagers is the same as the proportion of teenagers in the population, i.e. 0.07. The alternative hypothesis (\(H_a\)) is that the proportion of accidents involving teenagers is different from 0.07, hence the two tailed test. \nSo, \(H_0: P = 0.07\) and \(H_a: P \neq 0.07\).

Calculate the Test Statistic

The test statistic z can be calculated using the formula:\[z = \frac{(P - P_0) \sqrt {n}}{\sqrt{P_0(1 - P_0)}}\]Where in this case, \(P = 0.14\) (from the sample), \(P_0 = 0.07\) (from the population), and \(n = 500\) (sample size).

Compute the p-value

The p-value is found by looking up the absolute value of test statistic z in the standard normal (z) table or calculator. Remember it’s a two-tailed test, so the p-value is two times the value found on the table.

Reject or Fail to Reject the Null Hypothesis

If the p-value is smaller than the significance level (\( \alpha \)), in general, a 0.05 level is used, we reject the null hypothesis. If it’s larger, we fail to reject the null hypothesis.