Question

9. As part of a study of corporate employees, the director of human resources for PNC Inc. wants to compare the distance traveled to work by employees at its office in downtown Cincinnati with the distance for those in downtown Pittsburgh. A sample of 35 Cincinnati employees showed they travel a mean of 370 miles per month. A sample of 40 Pittsburgh employees showed they travel a mean of 380 miles per month. The population standard deviations for the Cincinnati and Pittsburgh employees are 30 and 26 miles, respectively. By following the six-step procedure for hypothesis testing found below, answer the following: At the 0.05 significance level, is there a difference in the mean number of miles traveled per month between Cincinnati and Pittsburgh employees?

Step 1: State the Null Hypothesis (H_0) and the Alternate Hypothesis (H_1)

Step 2: Determine the level of significance. (Note: It’s given in this problem!)

Step 3: Select the Test Statistic

Step 4: Formulate the Decision Rule

Step 5: Make a Decision

Step 6: Interpret the Result

Answer 1

Let be the the mean number of miles traveled by an employee in Cincinnati.

Let be the the mean number of miles traveled by an employee in Pittsburgh.

Given:

For Cincinnati: = 370, s₁ = 30, n₁ = 35

For Pittsburgh: = 380, s₂ = 26, n₂ = 40

Since s₁/s₂ = 30/26 = 1.011 (it lies between 0.5 and 2) we used the pooled variance

The degrees of freedom used is n1 + n2 - 2 = 35 + 40 -2 = 73 (since pooled variance is used)

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Step 1:The Hypothesis:

H0:

Ha:

This is a Two tailed test.

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Step 2: The Level of significance = 0.05

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Step 3: The Test Statistic:   We use the students t test for 2 samples as population standard deviations are unknown.

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Step 4: The Decision Rule

If tobs is > tcritical, then Reject H0

Also, if p value is < , then reject H0.

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Step 5: The Decision:

The Critical Value:   The critical value (2 tail) at Alpha = ,df = 73, tcritical= +1.674 and -1.674

The p Value: The p value (2 Tail) for t = -2.11, df = 73, is; p value = 0.0397

Therefore t observed (-2.11) is < -tcritical, hence we reject H0.

Also p value (0.0397) is < 0.05, hence we reject H0.

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Step 6: The Conclusion: There is sufficient evidence at the 0.05 level to conclude that there is a difference in the mean number of miles traveled by employees in Cincinnati and employees in Pittsburgh.

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