Question

Seat belts help prevent injuries in automobile accidents, but they certainly don't offer complete protection in extreme situations. A random sample of 319 front-seat occupants involved in head-on collisions in a certain region resulted in 95 people who sustained no injuries ("Influencing Factors on the Injury Severity of Restrained Front Seat Occupants in Car-to-Car Head-on Collisions," Accident Analysis and Prevention \([1995]: 143-150\) ). Does this suggest that the true (population) proportion of uninjured occupants exceeds .25? State and test the relevant hypotheses using a significance level of \(.05\).

Short answer

The exact conclusion will depend on the computed P-value and the decision made in Step 4, but in general, the conclusion would be that there is sufficient/not sufficient evidence at the 0.05 level of significance to conclude that the population proportion of uninjured occupants in head-on collisions exceeds 0.25.

Step-by-step solution

State the Hypotheses

The null hypothesis, denoted by \(H_0\), is that the population proportion \(p\) is equal to 0.25. The alternative hypothesis, denoted by \(H_A\), is that the population proportion \(p\) is greater than 0.25.\nSo, \(H_0: p = 0.25\)\n \(H_A: p > 0.25\)

Compute the Test Statistic

The test statistic for testing a hypothesis about a population proportion \(p\) when the null hypothesis is \(H0: p = p0\) is \[Z = \frac{\hat{p} - p0}{\sqrt{\frac{p0(1 - p0)}{n}}}\] where \(\hat{p}\) is the sample proportion and \(n\) is the sample size. Here, \(\hat{p} = \frac{95}{319}\) and \(n = 319\). So, plug in values and compute the test statistic.

Find the P-value

The P-value is the probability that a standard normal random variable is more than the observed value of the test statistic, under the null hypothesis. You can find this by looking up the value of the test statistic in a standard normal table or by using technology.

Make a Decision

If the P-value is less than the significance level of 0.05, reject the null hypothesis. Otherwise, do not reject the null hypothesis.

State the Conclusion

Based on the decision in Step 4, state the conclusion in context: If the null hypothesis is rejected, there is sufficient evidence at the 0.05 level of significance to conclude that the population proportion of uninjured occupants in head-on collisions exceeds 0.25. If the null hypothesis is not rejected, there is not sufficient evidence at the 0.05 level to conclude that the population proportion exceeds 0.25.