A study was made of 413 children who were hospitalized as a result of motor vehicle crashes. Among 290 children who were not using seat belts, 50 were injured severely. Among children using seat belts, 16 were injured severely (based on data from “Morbidity Among Pediatric Motor Vehicle Crash Victims: The Effectiveness of Seat Belts,’’ by Osberg and Di Scala, American Journal of Public Health, Vol. 82, No. 3).
a) State the null hypothesis and alternative hypothesis.
b) Compute the test statistic.
c) Is there sufficient sample evidence to conclude, at the 0.05 significance level, that the rate of severe injuries is lower for children wearing seat belts?
d) Calculate the p-value.
e) State a conclusion about the null hypothesis and state a final conclusion that addresses the original claim
a) The null hypothesis (H0) can be stated as "The rate of severe injuries is the same for children wearing seat belts and those not wearing seat belts." The alternative hypothesis (Ha) can be stated as "The rate of severe injuries is lower for children wearing seat belts."
b) To compute the test statistic, we need to use a test for comparing two proportions. The test statistic for comparing two proportions is given by:
z = (p1 - p2) / √[(p-hat * (1 - p-hat) * (1/n1 + 1/n2)]
Where p-hat1 and p-hat2 are the sample proportions for severe injuries for children wearing seat belts and not wearing seat belts, n1 and n2 are the sample sizes, respectively.
In this case, p-hat1 = 16/123 (16 severe injuries out of 123 children wearing seat belts) and p-hat2 = 50/290 (50 severe injuries out of 290 children not wearing seat belts). The sample sizes are n1 = 123 and n2 = 290.
c) To determine if there is sufficient sample evidence to conclude that the rate of severe injuries is lower for children wearing seat belts, we need to compare the calculated test statistic to the critical value of the test.
We assume a significance level of 0.05. Using a standard normal distribution, the critical value for a one-tailed test at a 0.05 level of significance is approximately 1.645.
If the calculated test statistic is less than the critical value, we can reject the null hypothesis in favor of the alternative hypothesis.
d) The p-value is the probability of observing a test statistic as extreme as the one calculated (or more extreme), assuming the null hypothesis is true. To calculate the p-value, we can use the standard normal distribution and find the area under the curve corresponding to the calculated test statistic.
e) Based on the p-value, we can make a conclusion about the null hypothesis. If the p-value is less than the chosen significance level (0.05 in this case), we can reject the null hypothesis. The final conclusion will address the original claim, which is whether the rate of severe injuries is lower for children wearing seat belts.