Question

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.

Answer 1

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.