Math Question help

Question

The local police force gives all applicants an entrance exam and accepts only those applicants who score in the top 15% in this test. If the mean is score this year is 87 and the standard deviatiion is 8 would an individual with a score of 110 be accepted?

Answer 1

Use Chebyshev's Inequality. Chebyshev's inequality states that for any distribution (does not even have to be a normal one), the proportion of observations outside k standard deviations of the mean is at most

1/k².

We can use a handy measure called a z-score to figure out how many standard deviations 110 is from the mean of 87. That calculation is

(110 - 87)/8 = 2.875.

OK! That looks pretty high. Let's see if the officer makes the upper 15th percentile.

1/k² = 1/2.875² = 0.121

In other words, at most 12.1% of the population of officer test scores is farther than 2.875 standard deviations away from the mean. So, at worst, 110 is at the 100% - 12.1% or 87.9% percentile. He makes the 85 percentile requirement.