Final answer:
The hypothesis test comparing state and federal employee salaries at a 0.05 significance level indicates that state employees earn less on average, as the test statistic, -2.407, is less than the critical value of -1.645.
Step-by-step explanation:
Hypothesis Testing for the Salaries of State and Federal Employees
When conducting a hypothesis test to compare the mean salary of state employees to that of federal employees, we perform the following steps:
- State the hypotheses: The null hypothesis (H0) is that the mean salary of state employees, μ, is equal to or greater than that of federal employees. Formally, H0: μ ≥ $59,593. The alternative hypothesis (Ha), which is also the claim, is that the mean salary of state employees is less than that of federal employees, Ha: μ < $59,593.
- Find the critical value(s): For a one-tailed test at the 0.05 level of significance, the critical z-value is -1.645.
- Compute the test value: The test statistic z is calculated using the formula
z = (μ - μ0) / (σ / √ n), where μ0 is the mean salary of federal employees ($59,593), μ is the mean salary of state employees ($59,030), σ is the standard deviation ($1,500), and n is the sample size (35). This yields z = ($59,030 - $59,593) / ($1,500 / √ 35) = -2.407. - Make the decision: Since the test statistic, -2.407, is less than the critical value, -1.645, we reject the null hypothesis.
- Summarize the results: At a 0.05 level of significance, we have enough evidence to conclude that state employees earn on average less than federal employees.