Solution:

a) This response ignores sampling variability completely. It’s not good enough to compare 190 to 180, because we need to consider the fact that the sample mean distance will vary from sample to sample. The key issue is how much variability there is in the sample mean distances (how far they tend to fall from the population mean from sampling variability alone), and whether a sample mean distance of 190 yards is surprising.

b) The response confuses the standard deviation of the distances with the standard deviation of the sample means. The standard deviation of the sample means is / = 20/ = 2.5. Thus, the observed sample mean of 190 is (190-180)/2.5 = 4 standard deviations above the population mean of 180. Such a result would be extremely unlikely, so the observed sample mean does provide strong evidence that the population mean distance with the new ball is higher than 180.

c) This response ignores the fact that with a sample size as large as n=64, the Central Limit Theorem establishes that the sampling distribution of the sample mean will be approximately normal, and so the empirical rule applies, even if the population of distances is not normal.