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The past and recent time intervals (min) between eruptions of the Old Faithful geyser are listed in the table.

Null hypothesis:The past mean time interval between eruptions is equalto the recent mean time interval between eruptions.

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

Alternative hypothesis:The past mean time interval between eruptions is not equal to the recent mean time interval between eruptions.

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

The recent mean time interval between eruptions is equal to the following:

\(\begin{array}{c}{{\bar x}_1} = \frac{{\sum\limits_{i = 1}^{{n_1}} {{x_i}} }}{{{n_1}}}\\ = \frac{{78 + 91 + .... + 61}}{{17}}\\ = 78.82\end{array}\)

Therefore, the recent mean time interval between eruptions equals78.82 minutes.

The past mean time interval between eruptions equalsthe following:

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

Therefore, the past mean time interval between eruptions equals89.08 minutes.

The standard deviation of the recent time interval between eruptions is computed below:

\(\begin{array}{c}{s_1} = \sqrt {\frac{{\sum\limits_{i = 1}^{{n_1}} {{{({x_i} - {{\bar x}_1})}^2}} }}{{{n_1} - 1}}} \\ = \sqrt {\frac{{{{\left( {78 - 78.82} \right)}^2} + {{\left( {91 - 78.82} \right)}^2} + .... + {{\left( {61 - 78.82} \right)}^2}}}{{17 - 1}}} \\ = 13.97\end{array}\)

Therefore, the standard deviation of the recent time interval between eruptions equals 13.97 minutes.

The standard deviation for the past time interval between eruptions is equal to:

\(\begin{array}{c}{s_2} = \sqrt {\frac{{\sum\limits_{i = 1}^{{n_2}} {{{({x_i} - {{\bar x}_2})}^2}} }}{{{n_2} - 1}}} \\ = \sqrt {\frac{{{{\left( {89 - 89.08} \right)}^2} + {{\left( {88 - 89.08} \right)}^2} + .... + {{\left( {95 - 89.08} \right)}^2}}}{{12 - 1}}} \\ = 9.19\end{array}\)

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

Under the null hypothesis,\({\mu _1} - {\mu _2} = 0\).

The test statistic is computed below:

\(\begin{array}{c}t = \frac{{\left( {{{\bar x}_1} - {{\bar x}_2}} \right) - \left( {{\mu _1} - {\mu _2}} \right)}}{{\sqrt {\frac{{s_1^2}}{{{n_1}}} + \frac{{s_2^2}}{{{n_2}}}} }}\\ = \frac{{\left( {78.82 - 89.08} \right) - \left( 0 \right)}}{{\sqrt {\frac{{{{\left( {13.97} \right)}^2}}}{{17}} + \frac{{{{\left( {9.19} \right)}^2}}}{{12}}} }}\\ = - 2.385\end{array}\)

Thus, the value of the test statistic is -2.385.

Degrees of freedom: The smaller of the two values,\(\left( {{n_1} - 1} \right)\)and\(\left( {{n_2} - 1} \right)\)is considered as the degreesof freedom.

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

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

The value of the degrees of freedom is the minimum of (16,11) equal to 11.

Now see the t-distribution table for a two-tailed test with a 0.05 level of significance and 11 degrees of freedom.

The critical values are -2.201 and 2.201. The corresponding p-value is equal to 0.0362.

The value of the test statistic does not lie between the values -2.201 and 2.201 and the p-value is less than 0.05. Therefore, the null hypothesis is rejected at a 0.05 significance level.

There is sufficient evidence to support the claim that the mean time interval between eruptions has changed at a 0.05 level of significance.

Let the level of significance be equal to 0.01.

Referring to the t-distribution table, the critical values of t at\(\alpha = 0.01\) with 11 degrees of freedom for a two-tailed test are -3.1058 and 3.1058.

The p-value remains the same and is equal to 0.0362.

The test statistic value equal to -2.385 lies between the two critical values, and the p-value is greater than 0.01.

There is insufficient evidence to support the claim that the mean time interval between eruptions has changed at a 0.01 level of significance.

Thus, the conclusion of the claim changes with the change in the significance level.