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 54.0 inches and sigma equals 5 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

a) 34.46% of​ 10-year-old boys is tall enough to ride this​ coaster.

b) 78.81% of​ 10-year-old boys is tall enough to ride this​ coaster

c) 44.35% 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

Explanation:

When the distribution is normal, we use the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this question, we have that:

`\mu = 54, \sigma = 5`

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

This is 1 subtracted by the pvalue of Z when X = 56.

So

`Z = (X - \mu)/(\sigma)`

`Z = (56 - 54)/(5)`

`Z = 0.4`

`Z = 0.4` has a pvalue of 0.6554

1 - 0.6554 = 0.3446

34.46% of​ 10-year-old boys is tall enough to ride this​ 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?

This is 1 subtracted by the pvalue of Z when X = 50.

`Z = (X - \mu)/(\sigma)`

`Z = (50 - 54)/(5)`

`Z = -0.8`

`Z = -0.8` has a pvalue of 0.2119

1 - 0.2119 = 0.7881

78.81% 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?

Between 50 and 56 inches, which is the pvalue of Z when X = 56 subtracted by the pvalue of Z when X = 50.

From a), when X = 56, Z has a pvalue of 0.6554

From b), when X = 50, Z has a pvalue of 0.2119

0.6554 - 0.2119 = 0.4435

44.35% 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