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 between the two Z scores.
To find the probability of a rating that is 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 of 150 and 200 using the formula:
z = (x - μ) / σ
where
z is the z-score,
x is the given value,
μ is the mean, and
σ is the standard deviation.
For 150:
z1 = (150 - 175) / 15 = -25 / 15 ≈ -1.67
For 200:
z2 = (200 - 175) / 15 = 25 / 15 ≈ 1.67
Now, we need to find the area under the normal distribution curve between -1.67 and 1.67. This can be done using a standard normal distribution table or using statistical software.
Alternatively, we can use the cumulative distribution function (CDF) of the standard normal distribution:
P(-1.67 < Z < 1.67)
Using a standard normal distribution table or statistical software, we find that P(-1.67 < Z < 1.67) is approximately 0.8997.
Therefore, the probability of a rating that is between 150 and 200 is approximately 0.8997, or 89.97%.