a bank's loan officer rates applicants for credit. the ratings are normally distributed with a mean of 175 and a standard deviation of 15. if an applicant is randomly selected, find the probability of a rating that is between 150 and 200

Z = (score-mean)/SD

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion related to the Z scores.

To find the probability of a rating between 150 and 200, we need to calculate the area under the normal distribution curve between these two values.

First, let's standardize the values using the z-score formula:

z = (x - μ) / σ

Where:
z = the z-score
x = the value we want to standardize
μ = the mean of the distribution
σ = the standard deviation of the distribution

For the lower value (150):

z1 = (150 - 175) / 15 = -25 / 15 = -1.67

For the upper value (200):

z2 = (200 - 175) / 15 = 25 / 15 = 1.67

Next, we can use a standard normal distribution table or a calculator to find the probability associated with these z-scores. The area between -1.67 and 1.67 represents the probability of a rating between 150 and 200.

Using a standard normal distribution table, we find that the area to the left of -1.67 is approximately 0.0475 (or 4.75%), and the area to the left of 1.67 is also approximately 0.9525 (or 95.25%).

To find the area between these two z-scores, we subtract the smaller area from the larger area:

P(150 ≤ x ≤ 200) = 0.9525 - 0.0475 = 0.905

So, the probability of a rating between 150 and 200 is approximately 0.905, or 90.5%.