Dual Enrolled Statistics

Question

The distribution of heights of adult men in the U.S. is approximately normal with mean 69 inches and standard deviation 2.5 inches. Use what you know about a normal distribution and the 68-95-99.7 rule to answer the following.

a) About what percent of men are between 64 and 66.5 inches? 

b) Between what approximate heights do the middle 95 percent of men fall? 

c) Fill in the blank: About 2.5 percent of all men are shorter than ________. 

Answer 1

A) 13.59%. 66.5 inches is 1 standard deviation shorter than the mean, while 64 inches is 2 standard deviations from the mean. looking at a graph of the normal distribution, 13.59% of samples will fall between 1 and 2 standard deviations below the mean.

B) 66 to 72 inches. The middle 95% of men's hieght's fall within 2 standard deviation of the mean. The mean is 69 inches, and the stdev is 2.5 inches. Two standard deviations from the mean in either direction is 3 inches, so the range of heights is 3 inches less than 69, and 3 inches more than 69 inches.

C) 66 Inches. Since 95% of men are within 2 standard deviation of the mean in a normal distribution, 5% are not within 2 standard deviations. Half of that 5% are more than 2 standard deviations taller than the mean, and half are shorter than 2 standard deviations from the mean. Half of 5% is 2.5%, so 2.5% of men are more than 2 standard deviations shorter than the mean, or shorter than 66 inches.