To determine whether state employees earn less than federal employees at the 0.01 level of significance, we can perform a hypothesis test for the means. We are given the following information:
- Mean salary of federal employees (\(\mu_0\)): $59,593
- Mean salary of state employees (\(\bar{x}\)): $58,800
- Standard deviation of state salaries (\(s\)): $1,500
- Number of state employees (\(n\)): 30
Step 1: Set up the hypotheses.
- Null hypothesis (\(H_0\)): \(\mu = 59,593\) (state employees earn the same as federal employees)
- Alternative hypothesis (\(H_a\): \(\mu < 59,593\) (state employees earn less than federal employees)
Step 2: Calculate the test statistic.
To calculate the test statistic, we can use the formula for the z-test for the mean:
\[ z = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}} \]
Substituting in the values:
- \(\bar{x} = 58,800\)
- \(\mu_0 = 59,593\)
- \(s = 1,500\)
- \(n = 30\)
\[ z = \frac{58,800 - 59,593}{\frac{1500}{\sqrt{30}}} \]
Step 3: Calculate the denominator.
\[ \frac{1500}{\sqrt{30}} \approx \frac{1500}{5.477} \approx 273.86 \]
Step 4: Calculate the z-value.
\[ z = \frac{58,800 - 59,593}{273.86} = \frac{-793}{273.86} \approx -2.895 \]
Step 5: Round to the nearest hundredths.
Rounding \(-2.895\) to the nearest hundredths gives:
\[ \boxed{-2.90} \]
The test statistic value is \(-2.90\).