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?