Question

A civil service exam yields with a mean of 81 and a standard deviation of 5.5. Using Chebyshev's theorem, what can be said about the percentage of scores that fall within 2 standard deviations of the mean?

1) At least 75%
2) At least 88.9%
3) At least 93.8%
4) At least 96%

Answer 1

Chebyshev's theorem implies that at least 75% of scores from a civil service exam with a mean of 81 and a standard deviation of 5.5 will fall within 2 standard deviations of the mean.

Step-by-step explanation:

Chebyshev's Theorem and the Civil Service Exam

The student's question asks about the application of Chebyshev's theorem to determine the percentage of scores that fall within 2 standard deviations of the mean. Given that the mean score of a civil service exam is 81 with a standard deviation of 5.5, Chebyshev's theorem can be used to estimate the proportion of scores within a certain range.

Chebyshev's theorem states that for any data set, regardless of its distribution, at least (1 - (1/k^2)) of the data must fall within k standard deviations of the mean. Here, k is equal to 2, so we apply the theorem:

(1 - (1/2^2)) = (1 - 1/4) = (1 - 0.25) = 0.75

Therefore, at least 75% of the scores will fall within 2 standard deviations (±11 points) of the mean score of 81.