Question

In a distribution of 160 values with a mean of 72, least 120 fall within the interval 67–77. Approximately what percentage of values should fall in the interval 62–82? Use Chebyshev's theorem.

Answer 1

Using Chebyshev's theorem, at least 75% of the values should fall within 2 standard deviations of the mean in a given distribution, which in this case, translates to the interval 62–82.

Step-by-step explanation:

In a distribution of 160 values with a mean of 72, at least 120 fall within the interval 67–77. We can use Chebyshev's theorem to estimate the percentage of values that should fall within other intervals.

Chebyshev's theorem states that for any number k, where k is greater than 1, the proportion of values lying within k standard deviations of the mean is at least (1 - 1/k2). The interval 67–77 is 5 units away from the mean (72), so this is within approximately 1 standard deviation (because 120/160 = 75% and (1 - 1/12) = 0).

The interval 62–82 is 10 units away from the mean, or approximately 2 standard deviations. Using Chebyshev's theorem, at least (1 - 1/22) = 75% of the values fall within 2 standard deviations. As 75% is the minimum we can expect according to the theorem; in practice, this percentage might be even higher.