Use the Empirical Rule to Describe Data That Are Bell Shaped
-At a tennis tournament a statistician keeps track of every serve. The statistician reported that the mean serve speed of a particular player was 103 miles per hour (mph) and the standard deviation of the serve speeds was 9 mph. Assume that the statistician also gave us the information that the distribution of the serve speeds was bell shaped. What proportion of the player s serves are expected to be between 112 mph and 130 mph
A) $0.1585$
B) $0.317$
C) $0.997$
D) $0.68$

Question

Use the Empirical Rule to Describe Data That Are Bell Shaped
-At a tennis tournament a statistician keeps track of every serve. The statistician reported that the mean serve speed of a particular player was 103 miles per hour (mph) and the standard deviation of the serve speeds was 9 mph. Assume that the statistician also gave us the information that the distribution of the serve speeds was bell shaped. What proportion of the player s serves are expected to be between 112 mph and 130 mph 
A) $0.1585$
B) $0.317$
C) $0.997$
D) $0.68$

Quizplus · Accepted Answer

The Empirical Rule states that for a bell-shaped distribution, approximately 68% of the data falls within one standard deviation of the mean, 95% within two standard deviations, and 99.7% within three standard deviations. Here, one standard deviation from the mean (103 mph) is 103 ± 9 = [94, 112]. Two standard deviations is 103 ± 18 = [85, 121]. Three standard deviations is 103 ± 27 = [76, 130]. The question asks for the proportion between 112 mph and 130 mph, which is beyond one standard deviation but within three. Since 95% is within two standard deviations and 99.7% is within three, the proportion between the second and third standard deviation (both sides combined) is 99.7% - 95% = 4.7%. Since we're only looking at one side (above the mean), we take half of 4.7%, which is approximately 2.35%. However, this also includes the range from 94 to 112 mph, which is within one standard deviation and accounts for 34% of the data (half of 68%). Therefore, the correct proportion is less than 2.35% but more than what would be expected by just looking at the standard deviations directly. The closest choice given is 0.1585 or 15.85%, which is a misunderstanding of the Empirical Rule's application in this context. The correct calculation involves more precise statistical methods beyond the basic Empirical Rule, but none of the provided choices accurately reflect the correct proportion based on the Empirical Rule's direct application. The explanation provided is to clarify the misunderstanding; however, based on the Empirical Rule alone, none of the choices accurately capture the proportion for the range specified without more detailed statistical calculations.

Use the Empirical Rule to Describe Data That Are Bell 0.15850.15850.1585

Use the Empirical Rule to Describe Data That Are Bell $0.1585$