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?

Which of the following is a correct conclusion:

a) There is sufficient evidence to warrant rejection of the claim that state employees earn on average less than federal employees.

b) There is not sufficient evidence to warrant rejection of the claim that state employees earn on average less than federal employees.

c) The sample data support the claim that state employees earn on average less than federal employees

d) There is not sufficient sample evidence to support the claim that state employees earn on average less than federal employees.

Answer 1

To determine whether the average salary of state employees is less than that of federal employees, we can perform a hypothesis test.

Let's set up the hypotheses:

- Null Hypothesis (H0): μ_state >= μ_federal (State employees earn on average at least as much as federal employees)
- Alternative Hypothesis (H1): μ_state < μ_federal (State employees earn on average less than federal employees)

Where:
- μ_state = mean salary of state employees
- μ_federal = mean salary of federal employees ($59,593)

Given the information:
- Sample mean of state employees (x̄) = $58,800
- Federal employees mean (μ_federal) = $59,593
- Sample size (n) = 30
- Standard deviation (σ) of state employees = $1,500
- Significance level (α) = 0.01

Next, we need to calculate the test statistic using the formula for the z-test for a single sample mean:

\[
z = \frac{x̄ - μ_federal}{σ / \sqrt{n}}
\]

Substituting the known values:

\[
z = \frac{58,800 - 59,593}{1,500 / \sqrt{30}}
\]

Calculating the standard deviation of the sample mean (σ / √n):

\[
σ / \sqrt{n} = \frac{1,500}{\sqrt{30}} \approx 274.07
\]

Now, calculate the z-value:

\[
z = \frac{58,800 - 59,593}{274.07} \approx \frac{-793}{274.07} \approx -2.89
\]

Next, we need to find the critical z-value for a one-tailed test at the 0.01 significance level. This can be found using a z-table or calculator:

The critical z-value for α = 0.01 (one-tailed) is approximately -2.33.

Now, we can compare our calculated z-value with the critical z-value:

- Calculated z-value: -2.89
- Critical z-value: -2.33

Since -2.89 is less than -2.33, we reject the null hypothesis.

Based on our results:

- Rejecting the null hypothesis suggests there is sufficient evidence to support the alternative hypothesis that state employees earn on average less than federal employees.

Thus, the correct conclusion is:

**c) The sample data support the claim that state employees earn on average less than federal employees.**