To determine if state employees earn on average less than federal employees, we can use a one-sample t-test for the mean in this scenario since we're comparing the mean salary of state employees to a known population mean (the federal government's mean salary).
Given Information:
- Population mean salary of federal employees (μ) = $59,593
- Sample mean salary of state employees (\( \bar{x} \)) = $58,800
- Sample standard deviation (s) = $1,500
- Sample size (n) = 30
- Significance level (α) = 0.01
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)
Test Statistic:
Since we have the sample mean, sample standard deviation, and sample size, we can calculate the t-statistic using the formula:
\[ t = \frac{\bar{x} - \mu}{s / \sqrt{n}} \]
Where:
- \( \bar{x} \) = sample mean
- \( \mu \) = population mean
- \( s \) = sample standard deviation
- \( n \) = sample size
Plugging in the Values:
\[ t = \frac{58,800 - 59,593}{1,500 / \sqrt{30}} \]
First, calculate the standard error (SE):
\[ SE = \frac{s}{\sqrt{n}} = \frac{1500}{\sqrt{30}} \approx \frac{1500}{5.477} \approx 273.86 \]
Now calculate the t-statistic:
\[ t = \frac{-793}{273.86} \approx -2.90 \]
Degrees of Freedom:
The degrees of freedom (df) for this test is \( n - 1 \): \[ df = 30 - 1 = 29 \]
Critical Value:
For a one-tailed t-test at the 0.01 significance level with 29 degrees of freedom, we can look up the t-table or use a calculator.
The critical t-value (for one-tailed) at 0.01 significance level and df = 29 is approximately:
\[ t_{critical} \approx -2.462 \]
Decision Rule:
- If \( t < t_{critical} \), we reject the null hypothesis.
- If \( t \geq t_{critical} \), we fail to reject the null hypothesis.
Conclusion:
Since our calculated t-value (-2.90) is less than the critical value (-2.462), we reject the null hypothesis.
Final Conclusion:
At the 0.01 level of significance, there is sufficient evidence to conclude that state employees earn, on average, less than federal government employees.