Answer:
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
Step-by-step explanation:
When the distribution is normal, we use the z-score formula.
In a set with mean
and standard deviation
, the zscore of a measure X is given by:
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
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](https://tex.z-dn.net/?f=%5Cmu%20%3D%2054%2C%20%5Csigma%20%3D%205)
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 = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![Z = \frac{56 - 54}{5}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B56%20-%2054%7D%7B5%7D)
![Z = 0.4](https://tex.z-dn.net/?f=Z%20%3D%200.4)
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 = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![Z = \frac{50 - 54}{5}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B50%20-%2054%7D%7B5%7D)
![Z = -0.8](https://tex.z-dn.net/?f=Z%20%3D%20-0.8)
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