Answer:
a) 40.17% probability of a value between 75.0 and 90.0.
b) 35.94% probability of a value 75.0 or less.
c) 20.22% probability of a value between 55.0 and 70.0.
Step-by-step explanation:
Problems of normally distributed samples are solved using 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 problem, we have that:
![\mu = 80, \sigma = 14](https://tex.z-dn.net/?f=%5Cmu%20%3D%2080%2C%20%5Csigma%20%3D%2014)
a. Compute the probability of a value between 75.0 and 90.0.
This is the pvalue of Z when X = 90 subtracted by the pvalue of Z when X = 75.
X = 90
![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{90 - 80}{14}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B90%20-%2080%7D%7B14%7D)
![Z = 0.71](https://tex.z-dn.net/?f=Z%20%3D%200.71)
has a pvalue of 0.7611
X = 75
![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{75 - 80}{14}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B75%20-%2080%7D%7B14%7D)
![Z = -0.36](https://tex.z-dn.net/?f=Z%20%3D%20-0.36)
has a pvalue of 0.3594
0.7611 - 0.3594 = 0.4017
40.17% probability of a value between 75.0 and 90.0.
b. Compute the probability of a value 75.0 or less.
This is the pvalue of Z when X = 75. 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{75 - 80}{14}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B75%20-%2080%7D%7B14%7D)
![Z = -0.36](https://tex.z-dn.net/?f=Z%20%3D%20-0.36)
has a pvalue of 0.3594
35.94% probability of a value 75.0 or less.
c. Compute the probability of a value between 55.0 and 70.0.
This is the pvalue of Z when X = 70 subtracted by the pvalue of Z when X = 55.
X = 70
![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{70 - 80}{14}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B70%20-%2080%7D%7B14%7D)
![Z = -0.71](https://tex.z-dn.net/?f=Z%20%3D%20-0.71)
has a pvalue of 0.2389
X = 55
![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{55 - 80}{14}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7B55%20-%2080%7D%7B14%7D)
![Z = -1.79](https://tex.z-dn.net/?f=Z%20%3D%20-1.79)
has a pvalue of 0.0367
0.2389 - 0.0367 = 0.2022
20.22% probability of a value between 55.0 and 70.0.