Question

A bank's loan officer rates applicants for credit. the ratings are normally distributed with a mean of 200 and a standard deviation of 50. if an applicant is randomly selected, find the probability of a rating that is between 200 and 275.

Answer 1

The probability is 0.4332.

We want to find the z-scores associated with the ends of the ranges.  The z-score for 200 would be 0, since it is 0 away from the mean.  To find the z-score of 275 we use the formula:

$z=\frac{X-\mu}{\sigma} \\ \\z=\frac{275-200}{50}=\frac{75}{50}=1.5$

Using a z-table (http://www.z-table.com) we can find the area under the curve to the left of 1.5 (everything less than that, so every score less than 275).  This is 0.9338. However, we only want the range from the mean, 200, up to this point.  This means we subtract the area to the left of 200, 0.5 (half of the data goes from the mean left), from this:

0.9338-0.5=0.4338.