Answer :
Answer:
a) The credit score that defines the upper 20% is 705.
b) Eighty-five percent of the customers will have a credit score higher than 470.38.
Step-by-step explanation:
Normal Probability Distribution:
Problems of normal distributions can be solved using the z-score formula.
In a set with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the zscore of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
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.
Mean of 600 and a standard deviation of 125.
This means that [tex]\mu = 600, \sigma = 125[/tex]
(a) Find the credit score that defines the upper 20 percent.
This is the 100 - 20 = 80th percetile, which is X when Z has a pvalue of 0.8. So X when Z = 0.84.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]0.84 = \frac{X - 600}{125}[/tex]
[tex]X - 600 = 0.84*125[/tex]
[tex]X = 705[/tex]
The credit score that defines the upper 20% is 705.
(b) Eighty-five percent of the customers will have a credit score higher than what value?
This is the 100 - 85 = 15th percentile, which is X when Z has a pvalue of 0.15. So X when Z = -1.037.
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]-1.037 = \frac{X - 600}{125}[/tex]
[tex]X - 600 = -1.037*125[/tex]
[tex]X = 470.38[/tex]
Eighty-five percent of the customers will have a credit score higher than 470.38.