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Two samples are considered. One sample represents the recent time intervals between eruptions with a sample size equal to 17,and the other represents past time intervals between eruptions with a sample size equal to 12.

It is claimed that the variation inthe recent eruption times is not equal to the variation in the pasteruption times.

Let\({\sigma _1}\)and\({\sigma _2}\)be the population standard deviationsof the recent eruption timesandthe past eruption timesrespectively.

Null hypothesis: The population standard deviation of the recent eruption timesis equal to the population standard deviation of the past eruption times.

Symbolically,

\({H_0}:{\sigma _1} = {\sigma _2}\)

Alternate Hypothesis: The population standard deviation of the recent eruption times is not equal to the population standard deviation of the past eruption times.

Symbolically,

\({H_1}:{\sigma _1} \ne {\sigma _2}\)

The sample variance has the following formula:

\({s^2} = \frac{1}{{n - 1}}\sum\limits_{i = 1}^n {{{\left( {x - \bar x} \right)}^2}} \)

The sample mean recent eruption time is equal to:

\(\begin{array}{c}{{\bar x}_1} = \frac{{78 + 91 + ....... + 61}}{{17}}\\ = 78.82\end{array}\)

The sample standard deviation of the recent eruption times is computed below:

\(\begin{array}{c}{s_{recent}} = \sqrt {\frac{{\sum\limits_{i = 1}^{{n_1}} {{{({x_i} - {{\bar x}_1})}^2}} }}{{{n_1} - 1}}} \\ = \sqrt {\frac{{{{\left( {78 - 78.82} \right)}^2} + {{\left( {91 - 78.82} \right)}^2} + .... + {{\left( {61 - 78.82} \right)}^2}}}{{17 - 1}}} \\ = 13.97\end{array}\)

Therefore, the standard deviation of the recent time interval between eruptions is equal to 13.97 minutes.

The sample mean past eruption time is equal to:

\(\begin{array}{c}{{\bar x}_2} = \frac{{\sum\limits_{i = 1}^{{n_2}} {{x_i}} }}{{{n_2}}}\\ = \frac{{89 + 88 + ... + 95}}{{12}}\\ = 89.08\end{array}\)

The sample standard deviation of the past eruption times is computed below:

\(\begin{array}{c}{s_{past}} = \sqrt {\frac{{\sum\limits_{i = 1}^{{n_2}} {{{({x_i} - {{\bar x}_2})}^2}} }}{{{n_2} - 1}}} \\ = \sqrt {\frac{{{{\left( {89 - 89.08} \right)}^2} + {{\left( {88 - 89.08} \right)}^2} + .... + {{\left( {95 - 89.08} \right)}^2}}}{{12 - 1}}} \\ = 9.19\end{array}\)

Therefore, the standard deviation of the past timeinterval between eruptions is equal to 9.19 minutes.

Since two independent samples involve a claim about the population standard deviation, apply an F-test.

Consider the larger sample variance to be\(s_1^2\)and the corresponding sample size to be\({n_1}\).

The following values are obtained:

\({\left( {13.97} \right)^2} = 195.1609\)

\({\left( {9.19} \right)^2} = 84.4561\)

Here,\(s_1^2\)is the sample variance corresponding to recent eruption times and has a value equal to 195.1609 minutes squared.

\(s_2^2\)is the sample variance corresponding to the past eruption times and has a value equal to 84.4561 minutes squared.

Substitute the respective values to calculate the F statistic:

\(\begin{array}{c}F = \frac{{s_1^2}}{{s_2^2}}\\ = \frac{{{{\left( {13.97} \right)}^2}}}{{{{\left( {9.19} \right)}^2}}}\\ = 2.311\end{array}\)

The value of the numerator degrees of freedom is equal to:

\(\begin{array}{c}{n_1} - 1 = 17 - 1\\ = 16\end{array}\)

The value of the denominator degrees of freedom is equal to:

\(\begin{array}{c}{n_2} - 1 = 12 - 1\\ = 11\end{array}\)

For the F test, the critical value corresponding to the right-tail is considered.

The critical value can be obtained using the F-distribution table with numerator degrees of freedom equal to 16and denominator degrees of freedom equal to 11 for a right-tailed test.

The level of significance is equal to:

\(\begin{array}{c}\frac{\alpha }{2} = \frac{{0.05}}{2}\\ = 0.025\end{array}\)

Thus, the critical value is equal to 3.3044.

The two-tailed p-value for F equal to 2.311 is equal to 0.1631.

Since the test statistic value is less than the critical value and the p-value is greater than 0.05, the null hypothesis fails to be rejected.

Thus, there is enough evidence to supportthe claimthat the variation of the times between eruptions has changed