Answer:
a) The credit score that defines the upper 5% is 764.50.
b) Seventy-five percent of the customers will have a credit score higher than 532.5.
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 = 600, \sigma = 100](https://tex.z-dn.net/?f=%5Cmu%20%3D%20600%2C%20%5Csigma%20%3D%20100)
(a) Find the credit score that defines the upper 5 percent.
Value of X when Z has a pvalue of 1-0.05 = 0.95. So X when Z = 1.645.
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![1.645 = \frac{X - 600}{100}](https://tex.z-dn.net/?f=1.645%20%3D%20%5Cfrac%7BX%20-%20600%7D%7B100%7D)
![X - 600 = 1.645*100](https://tex.z-dn.net/?f=X%20-%20600%20%3D%201.645%2A100)
![X = 764.5](https://tex.z-dn.net/?f=X%20%3D%20764.5)
The credit score that defines the upper 5% is 764.50.
(b) Seventy-five percent of the customers will have a credit score higher than what value
100 - 75 = 25
This the 25th percentile, which is the value of X when Z has a pvalue of 0.25. So it ix X when Z = -0.675.
![Z = \frac{X - \mu}{\sigma}](https://tex.z-dn.net/?f=Z%20%3D%20%5Cfrac%7BX%20-%20%5Cmu%7D%7B%5Csigma%7D)
![-0.675= \frac{X - 600}{100}](https://tex.z-dn.net/?f=-0.675%3D%20%5Cfrac%7BX%20-%20600%7D%7B100%7D)
![X - 600 = -0.675*100](https://tex.z-dn.net/?f=X%20-%20600%20%3D%20-0.675%2A100)
![X = 532.5](https://tex.z-dn.net/?f=X%20%3D%20532.5)
Seventy-five percent of the customers will have a credit score higher than 532.5.