A data set lists earthquake depths. The summary statistics are $n=400, x=4.42 \mathrm{~km}, \mathrm{~s}=4.27 \mathrm{~km}$. Use a 0.01 significance level to test the claim of a seismologist that these earthquakes are from a population with a mean equal to 4.00. Assume that a simple random sample has been selected. Identify the null and alternative hypotheses, test statistic, P-value, and state the final conclusion that addresses the original claim.

What are the null and alternative hypotheses?
A.
\[
\begin{array}{l}
H_{0}: \mu=4.00 \mathrm{~km} \\
H_{1}: \mu \neq 4.00 \mathrm{~km}
\end{array}
\]
C.
\[
\begin{array}{l}
H_{0}: \mu=4.00 \mathrm{~km} \\
H_{1}: \mu> 4.00 \mathrm{~km}
\end{array}
\]
B.
\[
\begin{array}{l}
H_{0}: \mu \neq 4.00 \mathrm{~km} \\
H_{1}: \mu=4.00 \mathrm{~km}
\end{array}
\]
D.
\[
\begin{array}{l}
H_{0}: \mu=4.00 \mathrm{~km} \\
H_{1}: \mu< 4.00 \mathrm{~km}
\end{array}
\]

Determine the test statistic.
(Round to two decimal places as needed.)

Determine the P-value.
(Round to three decimal places as needed.)

State the final conclusion that addresses the original claim.
$\mathrm{H}_{0}$. There is evidence to conclude that the mean of the population of earthquake depths is $4.00 \mathrm{~km}$ correct.

Step 1 ：The null and alternative hypotheses are: \(H_0: \mu = 4.00\) km and \(H_1: \mu \neq 4.00\) km

Step 2 ：The test statistic is calculated using the formula for a z-score: \(z = \frac{(x - \mu)}{(s / \sqrt{n})}\)

Step 3 ：Substituting the given values into the formula gives: \(z = \frac{(4.42 - 4.00)}{(4.27 / \sqrt{400})} = \frac{0.42}{(4.27 / 20)} = \frac{0.42}{0.2135} = 1.97\)

Step 4 ：The P-value is the probability that we would observe a test statistic as extreme as the one we calculated, assuming the null hypothesis is true. For a two-tailed test, we find the area to the right of our test statistic and multiply by 2: \(P-value = 2 * P(Z > 1.97) = 2 * (1 - 0.975) = 0.050\)

Step 5 ：Finally, we compare our P-value to our significance level (\(\alpha = 0.01\)). If the P-value is less than \(\alpha\), we reject the null hypothesis. If the P-value is greater than \(\alpha\), we fail to reject the null hypothesis.

Step 6 ：Since our P-value (0.050) is greater than our significance level (0.01), we fail to reject the null hypothesis. Therefore, there is not enough evidence to conclude that the mean of the population of earthquake depths is not 4.00 km. The seismologist's claim that the mean is 4.00 km could be correct. The final conclusion is \(\boxed{\text{Fail to Reject } H_0}\)