Problem
Use technology and a t-test to test the claim about the population mean $\mu$ at the given level of significance $\alpha$ using the given sample statistics. Assume the population is normally distributed Claim: $\mu>76 ; \alpha=0.05$ Sample statistics: $\bar{x}=76.4, s=3.2, n=26$ What are the null and alternative hypotheses? Choose the correct answer below. A. \[ \begin{array}{l} H_{0} \mu=76 \\ H_{A} \mu \neq 76 \end{array} \] c. \[ \begin{array}{l} H_{0}: \mu \geq 76 \\ H_{A}: \mu<76 \end{array} \] B. \[ \begin{array}{l} H_{0}: \mu \leq 76 \\ H_{A}: \mu>76 \end{array} \] D. \[ \begin{array}{l} H_{0}: \mu \neq 76 \\ H_{A}: \mu=76 \end{array} \] What is the value of the standardized test statistic? The standardized test statistic is $\square$. (Round to two decimal places as needed.)
Solution
Step 1 ：The null and alternative hypotheses are given by: \[H_{0}: \mu \leq 76\] \[H_{A}: \mu>76\]
Step 2 ：The standardized test statistic is calculated as follows: \[(\text{sample mean} - \text{population mean}) / (\text{sample standard deviation} / \sqrt{\text{sample size}})\]
Step 3 ：Substituting the given values, we get: \[(76.4 - 76) / (3.2 / \sqrt{26}) = 0.6373774391991072\]
Step 4 ：Rounding to two decimal places, the value of the standardized test statistic is \(\boxed{0.64}\)
