Question: A researcher is evaluating the effectiveness of a new physical education program for elementary school children. The program is designed to reduce competition and increase individual self-esteem. A sample of n = 16 children is selected and the children are placed in the new program. After 3 months, each child is given a standardized self-esteem test. For the general population of elementary school children, the scores on the self-esteem test form a normal distribution with µ = 40 and σ = 8.

(a) If the researcher obtains a sample mean of M = 42, is this enough evidence to conclude that the program has a significant effect? Assume a two-tailed test

with \(\alpha \) = .05. (show computation)

(b) If the sample mean is M = 44 , is this enough to demonstrate a significant effect? Again, assume a two-tailed test with \(\alpha \) = .05 (show computation)

(c) Briefly explain why you reach different conclusions for part (a) and part (b).