Part 2 of 4
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A simple random sample of size $n=15$ is drawn from a population that is normally distributed. The sample mean is found to be $\bar{x}=26.7$ and the sample standard deviation is found to be $s=6.3$ Determine if the population mean is different from 25 at the $\alpha=0.01$ level of significance. Complete parts (a) through (d) below.
(a) Determine the null and alternative hypotheses.
\[
\begin{array}{l}
\mathrm{H}_{0}: \mu=25 \\
\mathrm{H}_{1} \quad \mu=25
\end{array}
\]
(b) Calculate the P-value

P-value $=\square$
(Round to three decimal places as needed)

Step 1 ：State the null and alternative hypotheses. The null hypothesis is that the population mean is equal to 25, and the alternative hypothesis is that the population mean is not equal to 25. So, we have \(H_{0}: \mu=25\) and \(H_{1}: \mu \neq 25\).

Step 2 ：Given that the sample size \(n = 15\), the sample mean \(\bar{x} = 26.7\), and the sample standard deviation \(s = 6.3\).

Step 3 ：Calculate the test statistic using the formula \(t = \frac{\bar{x} - \mu}{s/\sqrt{n}}\). Substituting the given values, we get \(t = 1.0450907442146995\).

Step 4 ：Calculate the P-value. The P-value is the probability of observing a test statistic as extreme as the one calculated, assuming the null hypothesis is true. The P-value is calculated to be 0.31369108027570203.

Step 5 ：Compare the P-value with the significance level \(\alpha = 0.01\). The P-value is 0.314, which is greater than the significance level of 0.01. This means that we do not have enough evidence to reject the null hypothesis.

Step 6 ：Conclude the hypothesis test. Since the P-value is greater than the significance level, we cannot reject the null hypothesis. Therefore, we cannot conclude that the population mean is different from 25.

Step 7 ：The final answer is: The P-value is \(\boxed{0.314}\).