Seventy new jobs are opening up at an automobile manufacturing plant, and 1000 applicants show up for the 70 positions. To select the best 70 from among the applicants, the company gives a test that covers mechanical skill, manual dexterity, and mathematical ability. The mean grade on this test turns out to be 60, and the scores have a standard deviation of 6. Can a person who scores 84 count on getting one of the jobs? [Hint: Use Chebyshev's theorem.] Assume that the distribution is symmetric about the mean

As explained below, given that the score of the person is among the 0.03125 fraction of the best applicants, he can count on getting one of the jobs.

Explanation:

The hint is to use Chebyshev's Theorem.

Chebyshev's Theorem applies to any data set, even if it is not bell-shaped.

Chebyshev's Theorem states that at least 1-1/k² of the data lie within k standard deviations of the mean.

For this sample you have:

mean: 60 standard deviation: 6 score: 84

The number of standard deviations that 84 is from the mean is:

k = (score - mean) / standar deviation k = (84 - 60) / 6 = 24 / 6 = 4

Thus, the score of the person is 4 standard deviations above the mean.

How good is that?

Chebyshev's Theorem states that at least 1-1/k² of the data lie within k standard deviations of the mean. For k = 4, that is:

1 - 1/4² = 1 - 1/16 = 0.9375

That means that half of 1 - 0.9375 are above k = 4: 0.03125

Then, 1 - 0.03125 are below k = 4: 0.96875

Since there are 70 positions and 1,000 aplicants, 70/1,000 = 0.07. The compnay should select the best 0.07 of the applicants.

Given that the score of the person is among the 0.03125 upper fraction of the applicants, this person can count of geting one of the jobs.