Question

To qualify for a police academy, applicants are given a test of physical fitness. The scores are normally distributed with a mean of 64 and a standard deviation of 15. If only the top 10% of the applicants are selected, find the cutoff score.

Answer 1

The cutoff score for the top 10% of police academy applicants is approximately 83.2, which is calculated using the z-score for the 90th percentile in a normal distribution.

Step-by-step explanation:

To find the cutoff score for the top 10% of applicants for a police academy, where the test scores are normally distributed with a mean of 64 and a standard deviation of 15, we'll use the concept of z-scores and the properties of the normal distribution curve.

The z-score corresponding to the top 10% (90th percentile) in a normal distribution can be found using a z-table or calculator. For the top 10%, the z-score is approximately 1.28. Using the z-score formula:

Z = (X - Mean) / Standard Deviation

We can solve for X (the cutoff score):

1.28 = (X - 64) / 15

X = 1.28(15) + 64

X ≈ 19.2 + 64

X ≈ 83.2

Therefore, the cutoff score to be among the top 10% of applicants is approximately 83.2.