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?

Find p-value. Round it to the nearest thousandths.

1 answer

To determine whether state employees earn on average less than federal employees using hypothesis testing, we can perform a one-sample t-test. Here are the steps involved:

Step 1: Define the Hypotheses

  • Null Hypothesis (\(H_0\)): The mean salary of state employees is greater than or equal to the mean salary of federal employees. \[ H_0: \mu \geq 59,593 \]
  • Alternative Hypothesis (\(H_a\)): The mean salary of state employees is less than the mean salary of federal employees. \[ H_a: \mu < 59,593 \]

Step 2: Set the Significance Level

The significance level (\(\alpha\)) is 0.01.

Step 3: Calculate the Test Statistic

Use the t-test formula for the mean:

\[ t = \frac{\overline{x} - \mu}{s / \sqrt{n}} \] Where:

  • \(\overline{x} = 58,800\) (mean salary of state employees)
  • \(\mu = 59,593\) (mean salary of federal employees)
  • \(s = 1500\) (standard deviation of state employees)
  • \(n = 30\) (number of state employees)

Calculating the test statistic:

\[ t = \frac{58,800 - 59,593}{1500 / \sqrt{30}} = \frac{-793}{1500 / 5.477} = \frac{-793}{273.861} \approx -2.895 \]

Step 4: Determine the Degrees of Freedom

The degrees of freedom (df) for our t-test will be: \[ df = n - 1 = 30 - 1 = 29 \]

Step 5: Find the p-value

Using the t-distribution table or a calculator, we find the p-value for \(t = -2.895\) with 29 degrees of freedom.

Using a t-distribution calculator, the corresponding p-value for \(t = -2.895\) is approximately 0.0024.

Conclusion

At the 0.01 level of significance, we compare the p-value to \(\alpha\):

  • Since the p-value (0.0024) is less than 0.01, we reject the null hypothesis.

Final Result

The p-value, rounded to the nearest thousandths, is: \[ \text{p-value} \approx 0.002 \]

Thus, we can conclude that there is significant evidence at the 0.01 level to indicate that state employees earn less than federal employees on average.