The completion times for a certain marathon race was 2.4 hours with a standard deviation of 0.5 hours. What can you determine about these data by using Chebyshev's Inequality with K = 2?
A) At least 75% of the completion times are between 1.4 hours and 3.4 hours.
B) No more than 75% of the completion times are between 1.4 hours and 3.4 hours.
C) At most 88.9% of the completion times are between 1.4 hours and 3.4 hours.
D) At least 88.9% of the completion times are between 1.4 hours and 3.4 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 values fall within $k$ standard deviations of the mean. With $k = 2$, $1 - \frac{1}{2^2} = 1 - \frac{1}{4} = \frac{3}{4} = 0.75$ or 75%. Therefore, at least 75% of the completion times are within 2 standard deviations (2 x 0.5 = 1.0 hour) of the mean (2.4 hours), which is between 1.4 hours (2.4 - 1.0) and 3.4 hours (2.4 + 1.0).

The Completion Times for a Certain Marathon Race Was 2