Problem
A union of restaurant and foodservice workers would like to estimate the mean hourly wage, $\mu$, of foodservice workers in the U.S. this year The mean hourly wage last year was $\$ 8.25$, and there is good reason to believe that this year's value is less than last year's. The union decides to do a statistical test to see if the value has indeed decreased. The union chooses a random sample of this year's wages, computes the mean of the sample to be $\$ 7.85$, and computes the standard deviation of the sample to be $\$ 1.25$. Based on this information, complete the parts below. (a) What are the null hypothesis $H_{0}$ and the alternative hypothesis $H_{1}$ that should be used for the test? \[ \begin{array}{l} H_{0}:[ \\ H_{1}:[ \end{array} \] (b) Suppose that the union decides to reject the null hypothesis. What sort of error might it be making? (Choose one) (c) Suppose the true mean hourly wage for foodservice workers in the U.S. this year is $\$ 8.25$. Fill in the blanks to describe a Type I error. A Type I error would be (Choose one) $\mathbf{\nabla}$ the hypothesis that $\mu$ is
Solution
Step 1 ：The null hypothesis $H_{0}$ is a statement of no effect or no difference. In this case, the null hypothesis would be that the mean hourly wage has not decreased from last year, i.e., the mean hourly wage is still $8.25.
Step 2 ：The alternative hypothesis $H_{1}$ is what you might believe to be true or hope to prove true. In this case, the alternative hypothesis would be that the mean hourly wage has decreased from last year, i.e., the mean hourly wage is less than $8.25.
Step 3 ：So, the null hypothesis $H_{0}$ and the alternative hypothesis $H_{1}$ that should be used for the test are: \[\begin{array}{l} H_{0}: \mu = \$8.25 \\ H_{1}: \mu < \$8.25 \end{array}\]
