At a tennis tournament a statistician keeps track of every serve. The statistician reported that the mean servespeed of a particular player was 104 miles per hour (mph) and the standard deviation of the serve speeds was 8mph. Assume that the statistician also gave us the information that the distribution of the serve speeds was bellshaped. What proportion of the playerʹs serves are expected to be between 112 mph and 120 mph?
A) 0.1350
B) 0.270
C) 0.95
D) 0.68

Question

Quizplus · Accepted Answer

Given the distribution is bell-shaped (normal distribution), we can use the empirical rule or z-scores to find the proportion of serves between 112 mph and 120 mph. The mean is 104 mph, and the standard deviation is 8 mph. - 112 mph is 1 standard deviation above the mean (104 + 8 = 112).- 120 mph is 2 standard deviations above the mean (104 + 16 = 120).According to the empirical rule (68-95-99.7 rule), about 68% of data falls within 1 standard deviation of the mean, 95% within 2 standard deviations, and 99.7% within 3 standard deviations. However, we are interested in the proportion between 1 and 2 standard deviations above the mean.The area between the mean and 1 standard deviation (either side) is approximately 34% of the data (half of 68%). The area between the mean and 2 standard deviations (either side) is approximately 47.5% of the data (half of 95%). Therefore, the proportion of serves between 112 mph and 120 mph, which is between 1 and 2 standard deviations above the mean, is the difference between these two percentages: 47.5% - 34% = 13.5% or 0.1350.

At a Tennis Tournament a Statistician Keeps Track of Every