Data on the numbers of hospital adimissions resulting from motor vehicle crashes are given below for Fridays on the 6 th of a month and Fridays on the following 13th of the same month. Assume that the paired sample data is a simple random sample and that the differences have a distribution that is approximately normal. Construct a $95 \%$ confidence interval estimate of the mean of the population of differences between hospital admissions. Use the confidence interva to test the claim that when the 13th day of a month falls on a Friday, the numbers of hospital admissions from motor vehicle crashes are not affected. Friday the 6 th Friday the 13th 4 15 5 11 12 10 9 13 11 12 In this example, $\mu_{\mathrm{d}}$ is the mean value of the differences $\mathrm{d}$ for the population of all pairs of data, where each individual difference $d$ is defined as the number of hospital admissions on Friday the 6 th minus the number of hospital admissions on Friday the 13 th. Find the $95 \%$ confidence interval. (Round to two decimal places as needed.) Based on the confidence interval, can one reject the claim that when the 13 th day of a month falls on a Friday, the numbers of hospital admissions from motor vehicle crashes are not affected? A. No, because the confidence interval includes zero. B. No, because the confidence interval does not include zero. C. Yes, because the confidence interval includes zero. D. Yes, because the confidence interval does not include zero.
Solution
Step 1 ：First, calculate the differences between the number of hospital admissions on Friday the 6th and Friday the 13th. The differences are -11, -6, 2, -4, -1.
Step 2 ：Next, calculate the mean of these differences. The mean difference is -4.0.
Step 3 ：Then, calculate the standard deviation of these differences. The standard deviation is approximately 4.95.
Step 4 ：Using the mean and standard deviation, construct a 95% confidence interval for the mean difference. The confidence interval is approximately (-10.15, 2.15).
Step 5 ：Since the confidence interval includes zero, we cannot reject the claim that the number of hospital admissions from motor vehicle crashes is not affected when the 13th day of a month falls on a Friday.
Step 6 ：Final Answer: \(\boxed{\text{A. No, because the confidence interval includes zero}}\)
