Question

15. The police academy accepts only applicants who score a 226 or higher on the written test. Assume the distribution of scores on the written test is approximately Normal with a mean score of 200. If the top 10% of applicants score at least 226, find the standard deviation of scores on the written test

Answer 1

To find the standard deviation of scores on the written test, you can use z-scores and the standard Normal distribution table.

You are given that the mean score (μ) is 200, and you want to find the standard deviation (σ). You also know that the top 10% of applicants score at least 226.

To find the z-score corresponding to the top 10% (90th percentile), you can use a standard Normal distribution table or calculator. The z-score for the 90th percentile is approximately 1.28 (rounded to two decimal places).

Now, you can use the formula for a z-score:

z = (X - μ) / σ

Where X is the score you want to find (in this case, 226), μ is the mean (200), and σ is the standard deviation (which we are trying to find).

So, plug in the values:

1.28 = (226 - 200) / σ

Now, solve for σ:

1.28σ = 26

σ = 26 / 1.28

σ ≈ 20.31

Rounding to a reasonable number of decimal places, the standard deviation of scores on the written test is approximately 20.31.

Hope this helps! :)