A data set lists earthquake depths. The summary statistics are [tex]$ n = 600, \bar{x} = 5.32 \, \text{km}, s = 4.28 \, \text{km} $[/tex]. 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 5.00 km. 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. [tex]$ H_0: \mu = 5.00 \, \text{km} $[/tex]
[tex]$ H_1: \mu \neq 5.00 \, \text{km} $[/tex]

B. [tex]$ H_0: \mu = 5.00 \, \text{km} $[/tex]
[tex]$ H_1: \mu = 5.00 \, \text{km} $[/tex]

C.
[tex]\[
\begin{aligned}
H_0: \mu & = 5.00 \, \text{km} \\
H_1: \mu & < 5.00 \, \text{km}
\end{aligned}
\][/tex]

D. [tex]$ H_0: \mu = 5.00 \, \text{km} $[/tex]
[tex]$ H_1: \mu > 5.00 \, \text{km} $[/tex]

To solve the problem, we start by noting the seismologist's claim: the earthquakes come from a population with a mean depth of 5.00 km. This claim is stated as an equality, so the null hypothesis should reflect that exact value. Since no direction (greater than or less than) is specified for any deviation from 5.00 km, the test is two-tailed.

Thus, we set up the hypotheses as follows:

[tex]$$
H_0: \mu = 5.00 \text{ km}
$$[/tex]

[tex]$$
H_1: \mu \neq 5.00 \text{ km}
$$[/tex]

This means we are testing whether the actual mean depth is different from 5.00 km, either greater than or less than 5.00 km.

In summary:

- Null hypothesis: [tex]$$H_0: \mu = 5.00 \text{ km}$$[/tex]
- Alternative hypothesis: [tex]$$H_1: \mu \neq 5.00 \text{ km}$$[/tex]