In this scenario, you are comparing the mean salary of a sample of state employees to a known population mean (federal government employees' salary). Since the sample size is relatively small (n = 30), and the population standard deviation is not known (only the sample standard deviation is provided), you should use the t distribution for hypothesis testing.
Here are the steps to conduct the hypothesis test:
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State the hypotheses:
- Null hypothesis (H0): μ_state ≥ μ_federal (The average salary of state employees is greater than or equal to that of federal employees.)
- Alternative hypothesis (H1): μ_state < μ_federal (The average salary of state employees is less than that of federal employees.)
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Given data:
- Population mean (μ_federal) = $59,593
- Sample mean (M) = $58,800
- Sample standard deviation (s) = $1,500
- Sample size (n) = 30
- Significance level (α) = 0.01
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Calculate the test statistic: You would use the t-test formula: \[ t = \frac{M - μ_federal}{\frac{s}{\sqrt{n}}} \] Substituting the values: \[ t = \frac{58,800 - 59,593}{\frac{1,500}{\sqrt{30}}} \]
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Find the critical t-value: You will need to look up the t-table for α = 0.01 and degrees of freedom (df = n - 1 = 29) to find the critical t-value.
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Make a decision: If the calculated t-statistic is less than the negative of the critical t-value, you reject the null hypothesis.
In conclusion, you should use the t distribution for this hypothesis testing.
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