Question

"In a certain distribution of numbers, the mean is 70 with a standard deviation of 5. According to Chebyshev's theorem, for any set of numbers, the fraction that will lie within k standard deviations of the mean (for k > 1) is at least 1 - 1/k^2. Use Chebyshev's theorem to calculate the probability that a number from this distribution is either less than 50 or more than 90. Round your answer to the nearest thousandth."

Answer 1

The probability that a number from the given distribution is either less than 50 or more than 90, according to Chebyshev's theorem, is approximately 0.938.

Step-by-step explanation:

Chebyshev's theorem states that for any set of numbers, the fraction that will lie within k standard deviations of the mean (for k > 1) is at least 1 - 1/k^2. In this case, we want to calculate the probability that a number from the given distribution is either less than 50 or more than 90. To do this, we can use the complement rule.

First, let's calculate the number of standard deviations that 50 is away from the mean. (50 - 70) / 5 = -4. Next, we can calculate the number of standard deviations that 90 is away from the mean. (90 - 70) / 5 = 4. According to Chebyshev's theorem, at least 1 - 1/k^2 of the data will lie within k standard deviations of the mean. So, the probability that a number is either less than 50 or more than 90 is 1 - 1/4^2 = 1 - 1/16 = 15/16 ≈ 0.938.