You wish to test the lollowing claim $\left(H_{N}\right)$ at a significance level of $\alpha=0.001$ \[ \begin{array}{l} H_{1} * \mu-604 \\ H_{e}: \mu \neq 60.4 \end{array} \] You believe the population is normally distributed, but you do not know the standard devation. You obtain a sample of size $n=61$ with mean $M=58.6$ and a standard deviation of $S D=17.2$. What is the test statistic for this sample? (Report answer accurate to three decimal places) test statistic $=$ What is the p-value for this sample? (Report answer accurate to four decimal places:) pralue $=$ The p-value is. less than (or iequal to) $\alpha$ greater than a This test statistic leads to a decision to. reject the null accept the null fail to reject the null As such, the final conclusion is that. There is sufflcient evidence to warrant rejection of the claim that the population mean is not equal to 60.4 . There is not sufficient evidence to warrant rejection of the claim that the population mean is not equal to 60.4 . The sample data support the claim that the population mean is not equal to 60.4 . There is not sufflient sample evidence to support the claim that the population mean is not equal to 60.4 .
Solution
Step 1 ：Calculate the test statistic using the formula \(t = \frac{M - \mu}{SD / \sqrt{n}}\). Substitute the given values into the formula: \(t = \frac{58.6 - 60.4}{17.2 / \sqrt{61}}\).
Step 2 ：Simplify the calculation to get \(t = \frac{-1.8}{17.2 / 7.81}\).
Step 3 ：Further simplify the calculation to get \(t = \frac{-1.8}{2.20}\).
Step 4 ：Calculate the final value of the test statistic to get \(t = -0.818\).
Step 5 ：Calculate the p-value, which is the probability of observing a test statistic as extreme as -0.818, assuming the null hypothesis is true. Since this is a two-tailed test, we need to find the probability of observing a test statistic as extreme as -0.818 in either tail of the distribution. Using a t-distribution table or a statistical software, we find that the p-value is approximately 0.4164.
Step 6 ：Compare the p-value to the significance level. The p-value (0.4164) is greater than the significance level (0.001).
Step 7 ：Since the p-value is greater than the significance level, we fail to reject the null hypothesis.
Step 8 ：There is not sufficient evidence to warrant rejection of the claim that the population mean is not equal to 60.4. Therefore, \(\boxed{\text{We fail to reject the null hypothesis}}\).
