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.
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