Problem
You wish to test the following claim $\left(H_{a}\right)$ at a significance level of $\alpha=0.10$. \[ \begin{array}{l} H_{o}: \mu=78.2 \\ H_{a}: \mu>78.2 \end{array} \] You believe the population is normally distributed and you know the standard deviation is $\sigma=10.5$. You obtain a sample mean of $M=81.1$ for a sample of size $n=62$. What is the test statistic for this sample? (Report answer accurate to three decimal places.) \[ \text { test statistic }= \] What is the p-value for this sample? (Report answer accurate to four decimal places.) \[ \mathrm{p} \text {-value }= \] The $p$-value is... - less than (or equal to) $\alpha$
Solution
Step 1 ：Calculate the test statistic using the formula \( z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}} \)
Step 2 ：Plug in the given values: \( \bar{x} = 81.1 \), \( \mu = 78.2 \), \( \sigma = 10.5 \), and \( n = 62 \)
Step 3 ：Compute the test statistic: \( z = \frac{81.1 - 78.2}{\frac{10.5}{\sqrt{62}}} \)
Step 4 ：The test statistic is \( \boxed{2.175} \)
