Psychological Skills of Dancers. The Athletic Coping Skills Inventory (ACSI) is a test to measure psychological skills believed to influence athletic performance. Researcher Ellinor Klockare studied and evaluated the effects of a psychological skills training program on dancers and its usefulness to them in “A Psychological Skills Training Program for Dancers” (The Swedish School of Sport and Health Sciences, Master’s Thesis, 2012−2014). The study found that dancers’ TOPS scores are (approximately) normally distributed with mean 21.48 points and standard deviation 3.02 points.

a. Use the technology of your choice to simulate 1000 samples of four dancers’ TOPS scores each.

b. Find the sample mean of each of the 1000 samples obtained in part (a).

c. Obtain the mean, the standard deviation, and a histogram of the 1000 sample means.

d. Theoretically, what are the mean, standard deviation, and distribution of all possible sample means for samples of size 4?

e. Compare your answers from parts

(c) and (d).