Question

A survey found that​ women's heights are normally distributed with mean 63.8 in. and standard deviation 2.4 in. The survey also found that​ men's heights are normally distributed with a mean 68.5 in. and standard deviation 2.7. Most of the live characters at an amusement park have height requirements with a minimum of 4 ft 9 in. and a maximum of 6 ft 2 in.

Find the percentage of women meeting the height requirement. b) find the percentage of men meeting the height requirement the percentage of men meeting the height requirement (round to two decimal places as needed ) c) If the height requirements are changed to exclude only the tallest 5% of men and the shortest 5% of women what are the new height requirements the new height requirements are at least ___ in. and at most ___ in.

Answer 1

(a)

Let X is a random variable shows the woman's height. Here X has normal distribution with parameters

The z-score for X = 4ft 9in or 57 inches is

The z-score for X = 6ft 2in or 74 inches is

So percentage of women meet the height requirements is

Answer: 99.77%

b)

Let Y is a random variable shows the man's height. Here Y has normal distribution with parameters

The z-score for Y = 4ft 9in or 57 inches is

The z-score for Y = 6ft 2in or 74 inches is

So percentage of women meet the height requirements is

Answer: 97.93%

c)

The z-score 1.645 has 0.05 area to its right.

The upper limit of new height is

The z-score -1.645 has 0.05 area to its left.

The lower limit of new height is

The new height requirements are at least 59.852 in and at most 72.942 in.