A data set lists earthquake depths. The summary statistics are:

- \( n = 400 \)
- \(\overline{x} = 6.86 \, \text{km} \)
- \( s = 4.37 \, \text{km} \)

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 \( 6.00 \, \text{km} \). 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.
\[ H_0: \mu = 6.00 \, \text{km} \]
\[ H_1: \mu \neq 6.00 \, \text{km} \]

B.
\[ H_0: \mu \neq 6.00 \, \text{km} \]
\[ H_1: \mu = 6.00 \, \text{km} \]

C.
\[ H_0: \mu = 6.00 \, \text{km} \]
\[ H_1: \mu > 6.00 \, \text{km} \]

D.
\[ H_0: \mu = 6.00 \, \text{km} \]
\[ H_1: \mu < 6.00 \, \text{km} \]

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.

Fail to reject \( H_0 \).

There is insufficient evidence to conclude that the mean of the population of earthquake depths is \( 6.00 \, \text{km} \).

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.

To start with the hypothesis;

The null and the alternative hypothesis can be computed as :

[tex]H_o: \mu = 6.00 \\ \\ H_1 : \mu \neq 6.00[/tex]

The test statistics for this two tailed test can be computed as:

[tex]z= \dfrac{\overline x - \mu}{\dfrac{\sigma}{\sqrt {n}}}[/tex]

[tex]z= \dfrac{6.86 - 6.00}{\dfrac{4.37}{\sqrt {400}}}[/tex]

[tex]z= \dfrac{0.86}{\dfrac{4.37}{20}}[/tex]

z = 3.936

degree of freedom = n - 1

degree of freedom = 400 - 1

degree of freedom = 399

At the level of significance ∝ = 0.01

P -value = 2 × (z < 3.936) since it is a two tailed test

P -value = 2 × ( 1 - P(z ≤ 3.936)

P -value = 2 × ( 1 -0.9999)

P -value = 2 × ( 0.0001)

P -value = 0.0002

Since the P-value is less than level of significance , we reject [tex]H_o[/tex] at level of significance 0.01

Conclusion: There is sufficient evidence to conclude that the original claim that the mean of the population of earthquake depths is 5.00 km.