Test the claim about the population mean, $\mu$, at the given level of significance using the given sample statistics. Claim: $\mu=50 ; \alpha=0.02 ; \sigma=3.26$. Sample statistics: $\bar{x}=48.9, n=55$ Identify the null and alternative hypotheses. Choose the correct answer below. A. \[ \begin{array}{l} H_{0}: \mu=50 \\ H_{a}: \mu \neq 50 \end{array} \] C. \[ \begin{array}{l} H_{0}: \mu=50 \\ H_{a}: \mu>50 \end{array} \] E. \[ \begin{array}{l} H_{0}: \mu>50 \\ H_{a}: \mu=50 \end{array} \] B. \[ \begin{array}{l} H_{0}: \mu \neq 50 \\ H_{a}: \mu=50 \end{array} \] D. \[ \begin{array}{l} H_{0}: \mu=50 \\ H_{a}: \mu<50 \end{array} \] F. \[ \begin{array}{l} H_{0}: \mu<50 \\ H_{a}: \mu=50 \end{array} \] Calculate the standardized test statistic. The standardized test statistic is $\square$. (Round to two decimal places as needed.) Determine the critical value(s). Select the correct choice below and fill in the answer box to complete your choice. (Round to two decimal places as needed.) A. The critical values are $\pm \square$. B. The critical value is $\square$. Determine the outcome and conclusion of the test. Choose the correct answer below. A. Reject $\mathrm{H}_{0}$. At the $2 \%$ significance level, there is enough evidence to support the claim. B. Reject $\mathrm{H}_{0}$. At the $2 \%$ significance level, there is enough evidence to reject the claim. C. Fail to reject $\mathrm{H}_{0}$. At the $2 \%$ significance level, there is not enough evidence to support the claim. D. Fail to reject $\mathrm{H}_{0}$. At the $2 \%$ significance level, there is not enough evidence to reject the claim.
Solution
Step 1 ：Identify the null and alternative hypotheses. The null hypothesis, $H_0$, is that $\mu = 50$. The alternative hypothesis, $H_a$, is that $\mu \neq 50$.
Step 2 ：Calculate the standardized test statistic using the formula: $Z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}$. Substituting the given values, we get: $Z = \frac{48.9 - 50}{3.26 / \sqrt{55}}$. This gives us $Z = -2.06$.
Step 3 ：Determine the critical value(s). Since we are conducting a two-tailed test, we need to find the critical values that correspond to the upper and lower 1% of the standard normal distribution. The critical values are approximately $\pm 2.33$.
Step 4 ：Determine the outcome and conclusion of the test. Since the standardized test statistic, $-2.06$, is not less than the lower critical value, $-2.33$, and not greater than the upper critical value, $2.33$, we fail to reject the null hypothesis. This means that at the 2% significance level, there is not enough evidence to support the claim that the population mean is not 50.
