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 100. (a) Find the credit score that defines the upper 5 percent. (Use Excel or Appendix C to calculate the z-value. Round your final answer to 2 decimal places.) Credit score 764.50 (b) Seventy-five percent of the customers will have a credit score higher than what value

Answer 1

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.

Explanation:

Problems of normally distributed samples are 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.

In this problem, we have that:

`\mu = 600, \sigma = 100`

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

`1.645 = (X - 600)/(100)`

`X - 600 = 1.645*100`

`X = 764.5`

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

`-0.675= (X - 600)/(100)`

`X - 600 = -0.675*100`

`X = 532.5`

Seventy-five percent of the customers will have a credit score higher than 532.5.