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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.

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Notice that

[tex]1.67=3.81-2\times1.07[/tex]
[tex]5.95=3.81+2\times1.07[/tex]

This means you are looking for

[tex]\mathbb P(1.67<X<5.95)=\mathbb P(-2\times1.07<X-3.81<2\times1.07)=\mathbb P(|X-\mu|<2\sigma)[/tex]

Chebyshev's theorem states that for any constant [tex]k>0[/tex],

[tex]\mathbb P(|X-\mu|<k\sigma)\ge1-\dfrac1{k^2}[/tex]

which means

[tex]\mathbb P(1.67<X<5.95)=\mathbb P(|X-\mu|<2\sigma)\ge1-\dfrac14=\dfrac34[/tex]

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.

The eruptions lasted between [tex]1.67[/tex] and [tex]5.95[/tex] minutes are [tex]24[/tex] eruptions.

What is Chebyshev’s rule?

The proportion of any distribution that lies within [tex]$k$[/tex] standard deviations of the mean is at least [tex]$1-\left( \frac{1}{{{k}^{2}}} \right)$[/tex], where [tex]$k$[/tex] is any positive number larger than [tex]$1$[/tex] .

It is given that,

Mean [tex]$=3.81$[/tex]

Standard deviation [tex]$1.07$[/tex]

Sample size [tex]$n=32$[/tex]

The standard deviations from the mean are[tex]$1.67$[/tex] and [tex]$5.95$[/tex].

So,

[tex]$\text{Mean - 2 Standard deviation = }3.81-2\left( 1.07 \right)$[/tex]

                                               [tex]$=1.67$[/tex]

[tex]$\text{Mean + 2 Standard deviation = }3.81+2\left( 1.07 \right)$[/tex]

                                               [tex]$=5.95$[/tex]

Using Chebyshev’s rule with [tex]$k=2$[/tex],

We know that at least,

[tex]$100\left( 1-\frac{1}{{{k}^{2}}} \right)percent=100\left( 1-\frac{1}{4} \right)%$[/tex]

                      [tex]=75 percent[/tex]

Is within [tex]$2$[/tex] standard deviations from the mean.

So,

The corresponding number of the eruptions is the percentage multiplied by the sample size [tex]$n$[/tex]is,

[tex]$75\times n=0.75\times 32$[/tex]

[tex]$=24$[/tex]

Hence, there are [tex]24[/tex] eruptions  lasted between [tex]1.67[/tex] and [tex]5.95[/tex] minutes.

Learn more about Chebyshev’s rule here,

https://brainly.com/question/2426476

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