Question

Tai Chi is often recommended as a way to improve balance and flexibility in the elderly. Below are

before-and-after flexibility ratings (on a 1 to 10 scale, 10 being most flexible) for a random sample of 8 men in their 80's who took Tai Chi lessons for six months.

Subject A B C D E F G H

Flexibility rating after Tai Chi 2 4 3 3 3 4 5 10

Flexibility rating before Tai Chi 1 2 1 2 1 4 2 10

(a) Explain why these are paired data.

(b) Calculate and interpret the mean difference.

(c) Researchers would like to know if the true mean difference (after-before) in flexibility rating for men in their 80's who take Tai Chi lessons for 6 months is greater than zero. The P-value of this test is 0.004. Interpret this value.

(d) If the result of this study is statistically significant, can you conclude that the difference in the mean flexibility rating was caused by the Tai Chi lessons? Why or why not?

Answer 1

Answer a. These are paired data because

1. The Same Sample of Men in their 80's are used for the experimental study

2. There exists one to one relationship between the data points, i.e. Each data point in one data set is related to one, and only one, data point in the other data set. Here, the flexibility rating after Tai Chi of 2 is related to the flexibility rating of 2 before Tai Chi and no other.

Answer b.

After Tai Chi	Before Tai Chi	Difference
2	1	1
4	2	2
3	1	2
3	2	1
3	1	2
4	4	0
5	2	3
10	10	0

The mean difference is positive and is different from 0.

Answer c.

P-value is the probability of obtaining test results at least as extreme as the results actually observed during the test, assuming that the null hypothesis is correct.

This P value indicates that if Tai Chi had no effect, we'd obtain the observed difference or more in 4% of studies due to random sampling error.

Thus, p-value of 0.004 states that there is very low chance of obtaining the test results at least as extreme as the observed test-statistics obtained using paired t-test.

Answer d.

Here we are testing H0: There is no significant difference in the true mean (after-before) in flexibility rating for men in their 80's who take Tai Chi lessons for 6 months i.e

Alternative Hypothesis H1: Tthe true mean difference (after-before) in flexibility rating for men in their 80's who take Tai Chi lessons for 6 months

Here, since p-value = 0.004 < (alpha=0.05, say) so we reject Ho at alpha=0.05 level of significance and conclude that Tthe true mean difference (after-before) in flexibility rating for men in their 80's who take Tai Chi lessons for 6 months