The completion times for a certain marathon race was 2.9 hours with a standard deviation of 0.5 hours. What can you determine about these data by using Chebyshev's Inequality with K = 3?
A)No more than 88.9% of the completion times are between 1.4 hours and 4.4 hours.
B)At most 75% of the completion times are between 1.4 hours and 4.4 hours.
C)At least 88.9% of the completion times are between 1.4 hours and 4.4 hours.
D)At least 75% of the completion times are between 1.4 hours and 4.4 hours.

Question

Quizplus · Accepted Answer

Chebyshev's Inequality states that for any dataset and any constant $k > 1$, at least $1 - \frac{1}{k^2}$ of the data within a dataset will fall within $k$ standard deviations from the mean. With $k = 3$, $1 - \frac{1}{3^2} = 1 - \frac{1}{9} = \frac{8}{9}$ or approximately 88.9%. Therefore, at least 88.9% of the completion times are within 3 standard deviations (1.5 hours) of the mean (2.9 hours), which calculates to the range between 1.4 hours (2.9 - 1.5) and 4.4 hours (2.9 + 1.5).

The Completion Times for a Certain Marathon Race Was 2