Problem
Use a f-lest 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: $\mu \geq 8100 ; \alpha=0.10$ Sample statistics: $\hat{x}=7800, s=470, n=25$ What are the nuil and alternative hypotheses? A. \[ \begin{array}{l} H_{0} \mu \leq 8100 \\ H_{0}: \mu>8100 \end{array} \] C. \[ \begin{array}{l} H_{0}: \mu=8100 \\ H_{8}: \mu \neq 8100 \end{array} \] B. \[ \begin{array}{l} H_{0}: \mu \neq 8100 \\ H_{a}: \mu=8100 \end{array} \] D. \[ \begin{array}{l} H_{0}: \mu \geq 8100 \\ H_{4}: \mu<8100 \end{array} \] 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: \[ \begin{array}{l} H_{0}: \mu \geq 8100 \\ H_{a}: \mu<8100 \end{array} \]
Step 2 ：The value of the standardized test statistic is calculated as (sample mean - population mean) / (sample standard deviation / sqrt(sample size)).
Step 3 ：Substitute the given values into the formula: \[ t = \frac{7800 - 8100}{470 / \sqrt{25}} \]
Step 4 ：Calculate the value of the standardized test statistic: \[ t = -3.1914893617021276 \]
Step 5 ：Round the value of the standardized test statistic to two decimal places: \[ t = \boxed{-3.19} \]
