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A sample is taken from the eruption of an old faithful geyser with a sample size of 250 with the claim that the population mean of the prediction error in predicting the eruption is equal to zero.

The null hypothesis\({H_0}\)represents the population mean of the prediction error equal to 0. The alternate hypothesis\({H_1}\)represents the population mean of the prediction error, which is not equal to 0.

Let\(\mu \)be the population mean of the prediction error.

State the null and alternate hypotheses.

\(\begin{array}{l}{H_0}:\mu = 0\\{H_1}:\mu \ne 0\end{array}\)

The test statistic and the p-value are represented by the symbols\(t\)and\({\rm{P - value}}\),respectively.

The test statistic and p-value are obtained from the second row and the third row of the given output, respectively. Also, the critical value is obtained from the second row, as stated below.

\(\begin{array}{c}t \approx - 8.7201\\{\rm{P - value}} \approx 0.0000\\{\rm{Critical}}\;t = \pm 1.9695\end{array}\).

Reject the null hypothesis when the absolute value of the observed test statistics is greater than the critical value. Otherwise, fail to reject the null hypothesis.

\(\begin{array}{c}\left| { - 8.7201} \right| = 8.7201\\ > 1.9695\\t > t\left( {{\rm{critical}}} \right)\end{array}\).

The absolute value of the observed test statistic is significantly larger than the critical value. This implies that there is sufficient evidence to reject the null hypothesis.

As the null hypothesis is rejected, it can be concluded that there is insufficient evidence to support the claim that the population mean of the prediction error is not equal to zero.

As the prediction errors are not equal to 0, the prediction of eruption for the next time is not accurate.