Question

The credit scores of 35-year-olds applying for a mortgage at Ulysses Mortgage Associates are normally distributed with a mean of 600 and a standard deviation of 125. (a) Find the credit score that defines the upper 20 percent. (Use Excel or Appendix C to calculate the z-value. Round your final answer to 2 decimal places.) Credit score (b) Eighty-five percent of the customers will have a credit score higher than what value

Answer 1

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.

Explanation:

Normal Probability Distribution:

Problems of normal distributions 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 = (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.

Mean of 600 and a standard deviation of 125.

This means that
`\mu = 600, \sigma = 125`

(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.

`Z = (X - \mu)/(\sigma)`

`0.84 = (X - 600)/(125)`

`X - 600 = 0.84*125`

`X = 705`

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.

`Z = (X - \mu)/(\sigma)`

`-1.037 = (X - 600)/(125)`

`X - 600 = -1.037*125`

`X = 470.38`

Eighty-five percent of the customers will have a credit score higher than 470.38.