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 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.)

Answer 1

Answer:

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 $\mu$ and standard deviation $\sigma$, the zscore of a measure X is given by:

$Z = \frac{X - \mu}{\sigma}$

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 problem, we have that:

$\mu = 53.5, \sigma = 5$

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.

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{56 - 53.5}{5}$

$Z = 0.5$

$Z = 0.5$ 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.

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{50 - 53.5}{5}$

$Z = -0.5$

$Z = -0.5$ 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

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{56 - 53.5}{5}$

$Z = 0.5$

$Z = 0.5$ has a pvalue of 0.6915

X = 50

$Z = \frac{X - \mu}{\sigma}$

$Z = \frac{50 - 53.5}{5}$

$Z = -0.5$

$Z = -0.5$ 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