Question

From a sample with n=32, the mean duration of a geyser's eruption is 3.81 minutes and the standard deviation is 1.07 minutes. Using Chebychev's Theorem, determine at least how many of the eruptions lasted between 1.67 and 5.95 minutes.

Answer 1

Notice that

$1.67=3.81-2\times1.07$
$5.95=3.81+2\times1.07$

This means you are looking for

$\mathbb P(1.67$

Chebyshev's theorem states that for any constant $k>0$,

$\mathbb P(|X-\mu|$

which means

$\mathbb P(1.67$

In other words, at least 75% of the geysers will have eruptions lasting between 1.67 and 5.95 minutes. Out of 32 geysers, then, one can expect at least 24 of them to have eruptions with between these durations.