43. A researcher in the field of sports biomechanics believes he has found the "ideal" running posture for sprinters. He selects a random sample of 12 high school track sprinters and records their best $200 \mathrm{~m}$ time (in seconds). Then he trains these sprinters over several days to run using his proposed "ideal" posture. In the days after the träining ends, these sprinters run several $200 \mathrm{~m}$ races, and the researcher records their best performance from this time period. The before-training and after-training times (in seconds) are given below. Use the sign test to test the claim that the proposed ideal running posture is effective (i.e. the median of the "before - after" differences is positive). Use a 0.05 level of significance.
$\begin{array}{ccc}\text { Sprinter } & \text { Before } & \text { After } \\ 1 & 22.1 & 22.3 \\ 2 & 23.4 & 22.8 \\ 3 & 25.7 & 24.9 \\ 4 & 20.6 & 20.6 \\ 5 & 26.4 & 26.1 \\ 6 & 28.9 & 26.7 \\ 7 & 24.9 & 25.2 \\ 8 & 24.3 & 24.5 \\ 9 & 25.1 & 24.7 \\ 10 & 25.8 & 25.4 \\ 11 & 23.1 & 23.4 \\ 12 & 22.7 & 22.2\end{array}$
Identify the value of the TEST STATISTIC used in a hypothesis test of the claim.
a. 4
b. 7
c. -0.603
d. -0.866
Answer
Final Answer: The value of the test statistic used in a hypothesis test of the claim is \(\boxed{7}\).
Step 1 ：The researcher has recorded the best 200m times for 12 high school track sprinters both before and after training them in his proposed 'ideal' running posture. The times are as follows: Before: 22.1, 23.4, 25.7, 20.6, 26.4, 28.9, 24.9, 24.3, 25.1, 25.8, 23.1, 22.7 After: 22.3, 22.8, 24.9, 20.6, 26.1, 26.7, 25.2, 24.5, 24.7, 25.4, 23.4, 22.2
Step 2 ：We are testing the claim that the proposed ideal running posture is effective, which means the median of the 'before - after' differences is positive.
Step 3 ：To find the test statistic for the sign test, we need to calculate the differences between the before and after times for each sprinter, and count the number of positive differences.
Step 4 ：The differences between the before and after times are: -0.2, 0.6, 0.8, 0, 0.3, 2.2, -0.3, -0.2, 0.4, 0.4, -0.3, 0.5
Step 5 ：The positive differences are: 0.6, 0.8, 0.3, 2.2, 0.4, 0.4, 0.5
Step 6 ：The test statistic for the sign test is the number of positive differences, which is 7.
Step 7 ：Final Answer: The value of the test statistic used in a hypothesis test of the claim is \(\boxed{7}\).
