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 \(\boxed{7}\).