A data set lists earthquake depths. The summary statistics are $n=400, \bar{x}=4.89 \mathrm{~km}, s=4.44 \mathrm{~km}$. 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 altemative hypotheses? A. \[ \begin{array}{l} H_{0}: \mu=4.00 \mathrm{~km} \\ H_{1}: \mu<4.00 \mathrm{~km} \end{array} \] C. \[ \begin{array}{l} H_{0}: \mu=4.00 \mathrm{~km} \\ H_{1}: \mu \neq 4.00 \mathrm{~km} \end{array} \] Determine the test statistic. (Round to two decimal places as needed.) B. \[ \begin{array}{l} H_{0}: \mu \neq 4.00 \mathrm{~km} \\ H_{1}: \mu=4.00 \mathrm{~km} \end{array} \] D. \[ \begin{array}{l} H_{0}: \mu=4.00 \mathrm{~km} \\ H_{1}: \mu>4.00 \mathrm{~km} \end{array} \]
Solution
Step 1 ：The null and alternative hypotheses are: C. \[ \begin{array}{l} H_{0}: \mu=4.00 \mathrm{~km} \\ H_{1}: \mu \neq 4.00 \mathrm{~km} \end{array} \] The null hypothesis (H0) is that the population mean is equal to 4.00 km. The alternative hypothesis (H1) is that the population mean is not equal to 4.00 km. To determine the test statistic, we use the formula for a one-sample t-test: t = (x̄ - μ) / (s / √n) where x̄ is the sample mean, μ is the population mean under the null hypothesis, s is the sample standard deviation, and n is the sample size. Substituting the given values: t = (4.89 - 4.00) / (4.44 / √400) = 0.89 / 0.222 = 4.01 So, the test statistic is t = 4.01. To find the P-value, we look up the t-value in a t-distribution table or use a statistical software. For a two-tailed test with 399 degrees of freedom (n - 1), the P-value associated with t = 4.01 is less than 0.01. Since the P-value is less than the significance level of 0.01, we reject the null hypothesis. Therefore, the final conclusion is that there is sufficient evidence at the 0.01 level of significance to support the seismologist's claim that the mean earthquake depth is not 4.00 km.
