The completion times for a certain marathon race was 3 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 most 88.9% of the completion times are between 2 hours and 4 hours.
B) At least 75% of the completion times are between 2 hours and 4 hours.
C) At least 88.9% of the completion times are between 2 hours and 4 hours.
D) No more than 75% of the completion times are between 2 hours and 4 hours.

Question

Quizplus · Accepted Answer

Chebyshev's Inequality states that for any dataset, regardless of distribution, at least $1 - \frac{1}{K^2}$ of the data lies within $K$ standard deviations from the mean. With $K = 2$, this gives $1 - \frac{1}{2^2} = 1 - \frac{1}{4} = \frac{3}{4}$ or 75%. Therefore, at least 75% of the completion times are within 2 standard deviations (1 hour) of the mean (3 hours), which corresponds to between 2 hours and 4 hours.

The Completion Times for a Certain Marathon Race Was 3 0.50.50.5

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