The mean salary of federal government employees on the General Schedule is $59,593. The average salary of 30 state employees who do similar work is $58,800 with

= $1500. At the 0.01 level of significance, can it be concluded that state employees earn on average less than federal employees?

The test is

1 answer

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.