In college basketball, a shot made from beyond a designated arc radiating about 20 feet from the basket is worth three points, instead of the usual two points given for shots made inside that arc. Over his career, University of Florida basketball player Lee Humphrey made 45% of his three-point attempts. In one game in his final season, he made only 3 of 12 three-point shots, leading a TV basketball analyst to announce that Humphrey was in a shooting slump.
a. Assuming Humphrey has a 45% chance of making any particular three-point shot, find the mean and standard deviation of the sampling distribution of the proportion of three-point shots he will make out of 12 shots.
b. How many standard deviations from the mean is this game’s result of making 3 of 12 three-point shots?
c. If Humphrey was actually not in a slump but still had a 45% chance of making any particular three-point shot, explain why it would not be especially surprising for him to make only 3 of 12 shots. Thus, this is not really evidence of a shooting slump.