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 53.5 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? a. The proportion of 10-year-old boys tall enough to ride the coaster is 0.3085. (Round to four decimal places as needed.) b. The proportion of 10 year-old-boys tall enough to ride the smaller coaster is 0.7580. (Round to four decimal places as needed.) c. The proportion of 10-year-old boys tall enough to ride the coaster in part b but not tall enough to ride the coaster in part a is nothing. (Round to four decimal places as needed.)

a) The proportion of 10-year-old boys tall enough to ride the coaster is 0.3085

b) The proportion of 10 year-old-boys tall enough to ride the smaller coaster is 0.6925

c) The proportion of 10-year-old boys tall enough to ride the coaster in part b but not tall enough to ride the coaster in part a is 0.3840

Step-by-step explanation:

Problems of normally distributed samples can be solved using the z-score formula.

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

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 problem, we have that:

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

Taller than 56 inches, which is 1 subtracted by the pvalue of Z when X = 56.

has a pvalue of 0.6915

1 - 0.6915 = 0.3085

The proportion of 10-year-old boys tall enough to ride the coaster is 0.3085

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?

Taller than 50 inches, which is 1 subtracted by the pvalue of Z when X = 50.

has a pvalue of 0.3075

1 - 0.3075 = 0.6925

The proportion of 10 year-old-boys tall enough to ride the smaller coaster is 0.6925

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.

X = 56

has a pvalue of 0.6915

X = 50

has a pvalue of 0.3075

0.6915 - 0.3075 = 0.3840

The proportion of 10-year-old boys tall enough to ride the coaster in part b but not tall enough to ride the coaster in part a is 0.3840

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