Question

A survey found that? women's heights are normally distributed with mean 62.7 in. and standard deviation of 3.4 in. The survey also found that? men's heights are normally distributed with mean of 69.3 in. and a standard deviation of 3.6 in. Most of the live characters employed at an amusement park have height requirements of a minimum of 55 in. and a maximum of 64 in. Complete parts? (a) and? (b) below.

a)Find the percentage of men meeting the height requirement. What does the result suggest about the genders of the people who are employed as characters at the amusement? park?

The percentage is ______?

b)if the height requirements are changed to exclude only the tallest? 50% of men and the shortest? 5% of? men, what are the new height? requirements?

The new height requirements are a minimum of __ inches and a maximum of __ inches?

Answer 1

a) Let X be the random variable denoting the height of men

and Y be the random variable denoting the height of women.

Thus, X ~ N(67.7, 3.8) i.e. (X - 67.7)/3.8 ~ N(0,1)

and Y ~ N(62.3, 2.1) i.e. (Y - 62.3)/2.1 ~ N(0,1)

The percentage of men meeting the height requirement

= P(55 < X < 64)

= P[(55 - 67.7)/3.8 < (X - 67.7)/3.8 < (64 - 67.7)/3.8]

= P[- 3.3421 < (X - 67.7)/3.8 < - 0.9737]

= (-0.9737) - (-3.3421) = 0.1651 - 0.0004 = 0.1647. (Ans).

[ is the cdf of N(0,1)]

The percentage of women meeting the height requirement

= P(55 < Y < 64)

= P[(55 - 62.3)/2.1 < (Y - 62.3)/2.1 < (64 - 62.3)/2.1]

= P[-3.4762 < (Y - 62.3)/2.1 < 0.8095]

= (0.8095) - (-3.4762) = 0.7909 - 0.0003 = 0.7906. (Ans).

Thus, we have mostly women in the amusement park.

b) Let the minimum height of the tallest 50% men be a.

Thus, P(X < a) = 0.5 i.e. P[(X - 67.7)/3.8 < (a - 67.7)/3.8] = 0.5

i.e. [(a - 67.7)/3.8] = 0.5 i.e. (a - 67.7)/3.8 = (0.5)

i.e. (a - 67.7)/3.8 = 0 i.e. a = 67.7.

Let the maximum height of shortest 5% of the men be b.

Thus, P(X < b) = 0.05 i.e.P[(X - 67.7)/3.8 < (b - 67.7)/3.8] = 0.05

i.e. [(b - 67.7)/3.8] = 0.05 i.e. (b - 67.7)/3.8 = (0.05)

i.e. (b - 67.7)/3.8 = - 1.645 i.e. b = 61.449.

The new height requirement are a minimum of 61.449 inches

and a maximum of 67.7 inches. (Ans).