38. Identify the value of the TEST STATISTIC used in a hypothesis test of the following claim and sample data: Claim: "Golfers who had one lesson with this instructor had an average increase of more than 15 yards in the distance of their drives." A random sample of 5 golfers who had attended a free trial lesson with this golf instructor was selected. Before the lesson, each of these golfers hit several drives from a tee, and the average distance for each golfer was recorded (in yards). This process was repeated after the lesson. Assume that the differences between the "before" and "after" distances are approximately normally distributed. Test the claim at the 0.1 significance level. $\begin{array}{cccccc}\text { Golfer } & \text { Aaron } & \text { Buddy } & \text { Chloe } & \text { David } & \text { Eric } \\ \text { Dist Before Lesson } & 201.0 & 195.1 & 186.4 & 236.5 & 250.4 \\ \text { Dist After Lesson } & 224.7 & 208.6 & 200.6 & 264.3 & 261.9\end{array}$ (2) a. 0.980 b. 1.096 c. 0.995 d. 1.113
Solution
Step 1 ：First, calculate the differences between the 'before' and 'after' distances for each golfer. The differences are \(23.7, 13.5, 14.2, 27.8, 11.5\).
Step 2 ：Next, calculate the mean of these differences. The mean difference is approximately \(18.14\).
Step 3 ：Then, calculate the standard deviation of these differences. The standard deviation is approximately \(7.17\).
Step 4 ：Finally, calculate the t-statistic. The t-statistic is the difference between the sample mean and the hypothesized mean, divided by the standard error of the mean. The standard error of the mean is the standard deviation of the sample divided by the square root of the sample size. The calculated t-statistic is approximately \(0.98\).
Step 5 ：The value of the test statistic used in the hypothesis test is \(\boxed{0.980}\).
