Question

A new roller coaster at an amusement park requires individuals to be at least​ 4' 8" ​(56 ​inches) tall to ride. It is estimated that the heights of​ 10-year-old boys are normally distributed with mu equals μ=54.5 inches and σ=4 inches.

a. What proportion of​ 10-year-old boys is tall enough to ride the​ coaster?

b. A smaller coaster has a height requirement of 50 inches to ride. What proportion of​ 10-year-old boys is tall enough to ride this​ coaster?

c. What proportion of​ 10-year-old boys is tall enough to ride the coaster in part b but not tall enough to ride the coaster in part​ a?

Answer 1

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The normal distribution parameters are given as:

Mean, Mu = 54.5 ft

Stdev, Sigma = 4 ft

We use the above and the standardization formula:

Z = (X-Mean)/Sigma to convert a raw score to a normalized score.

a. P(X>=56) = ?
Standardizing using above formula:
= 1-P(Z<(56-54.5)/4)
= 1-P(Z<0.375)
= 1-0.6462
= .3548

b. P(X<50) = ?
Standardizing using above formula:
P(Z<(50-54.5)/4)
= P(Z<-1.125)
= 0.1303

c. P(50<X<56)
Standardizing using above formula:
P(Z<(-1.125<Z<0.375)
= .6462-.1303
= 0.5159