Test the claim about the population mean, $\mu$, at the given level of significance using the given sample statistics. Claim: $\mu=50 ; \alpha=0.03 ; \sigma=3.68$. Sample statistics: $\bar{x}=48.5, n=62$ Identify the null and alternative hypotheses. Choose the correct answer below. A. \[ \begin{array}{l} H_{0}: \mu \neq 50 \\ H_{a}: \mu=50 \end{array} \] c. \[ \begin{array}{l} H_{0}: \mu=50 \\ H_{a}: \mu \neq 50 \end{array} \] E. \[ \begin{array}{l} \mathrm{H}_{0}: \mu<50 \\ \mathrm{H}_{\mathrm{a}}: \mu=50 \end{array} \] Calculate the standardized test statistic. The standardized test statistic is (Round to two decimal places as needed.) B. \[ \begin{array}{l} H_{0}: \mu=50 \\ H_{a}: \mu>50 \end{array} \] D. \[ \begin{array}{l} H_{0}: \mu>50 \\ H_{a}: \mu=50 \end{array} \] F. \[ \begin{array}{l} H_{0}: \mu=50 \\ H_{a}: \mu<50 \end{array} \]

Step 1 ：The null hypothesis is usually a statement of no effect or no difference and is the hypothesis that we assume to be true. The alternative hypothesis is what we would conclude if the null hypothesis is rejected. In this case, the claim is that the population mean, \(\mu\), is 50. So, the null hypothesis, \(H_0\), should be \(\mu=50\). The alternative hypothesis, \(H_a\), is the opposite of the null hypothesis. Since we are not given a specific direction (greater than or less than), the alternative hypothesis should be \(\mu \neq 50\). So, the null and alternative hypotheses are: \[\begin{array}{l} H_{0}: \mu=50 \\ H_{a}: \mu \neq 50 \end{array}\]

Step 2 ：The standardized test statistic (Z-score) can be calculated using the formula: \[Z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}\] where \(\bar{x}\) is the sample mean, \(\mu\) is the population mean, \(\sigma\) is the population standard deviation, and \(n\) is the sample size.

Step 3 ：Substitute the given values into the formula: \[Z = \frac{48.5 - 50}{3.68 / \sqrt{62}}\]

Step 4 ：After calculating, we find that the standardized test statistic is -3.21.

Step 5 ：So, the final answer is: The null and alternative hypotheses are: \[\begin{array}{l} H_{0}: \mu=50 \\ H_{a}: \mu \neq 50 \end{array}\] The standardized test statistic is \(\boxed{-3.21}\).