Question 1186081: A manager at the head office of a company is considering 3 branch office managers for promotion. Branch reports include records of sales volume per agent for each branch. A random sample of records was selected for agents at each branch. All branches are located in cities with similar demographics (per capita income, population etc.) Using the samples, the manager wants to see if there is a significant difference in performance of agents at the three branches. If there is a difference, the information will be used to help determine which branch manager to promote; otherwise it will not be included in the decision. (All values are in hundreds of thousands of dollars.) (15 points)
Branch Managed
Branch Managed
Branch Managed
by Harrison
by Dale
by Stevenson
7.2
8.8
6.9
6.4
10.7
8.7
10.1
11.1
10.5
11
9.8
11.4
9.9
10.6
For the single measurement problem, use an a = 0.01 level of significance. Conduct an appropriate hypothesis test and conclude whether to reject or not reject the claim that there is no difference among the agents at the different branches.
Answer by CPhill(1987)   (Show Source):
You can put this solution on YOUR website! Here's how to conduct a one-way ANOVA test to determine if there's a significant difference in agent performance among the three branches:
**1. State the Hypotheses:**
* **Null Hypothesis (H₀):** There is no significant difference in the mean sales volume per agent among the three branches. (μ₁ = μ₂ = μ₃)
* **Alternative Hypothesis (H₁):** There is a significant difference in the mean sales volume per agent among at least two of the branches.
**2. Calculate the Sum of Squares:**
First, calculate the means for each branch and the overall mean:
* Harrison Mean (x̄₁) = (7.2 + 6.4 + 10.1 + 11 + 9.9) / 5 = 8.92
* Dale Mean (x̄₂) = (8.8 + 10.7 + 11.1 + 9.8 + 10.6) / 5 = 10.2
* Stevenson Mean (x̄₃) = (6.9 + 8.7 + 10.5 + 11.4) / 4 = 9.4
* Overall Mean (x̄) = (Sum of all values) / (Total number of values) = 9.42
Next, calculate the Sum of Squares Between Groups (SSB) and Sum of Squares Within Groups (SSW):
* SSB = 5(8.92 - 9.42)² + 5(10.2 - 9.42)² + 4(9.4 - 9.42)² = 1.25 + 3.136 + 0.0016 = 4.3876
* SSW = (7.2-8.92)²+(6.4-8.92)²+(10.1-8.92)²+(11-8.92)²+(9.9-8.92)² + (8.8-10.2)²+(10.7-10.2)²+(11.1-10.2)²+(9.8-10.2)²+(10.6-10.2)² + (6.9-9.4)²+(8.7-9.4)²+(10.5-9.4)²+(11.4-9.4)² = 2.9528+6.3504+1.3924+4.3264+0.9604+1.96+0.25+0.81+0.16+0.16+6.25+0.49+1.21+4 = 26.602
* Total Sum of Squares (SST) = SSB + SSW = 4.3876+26.602=30.9896
**3. Calculate the Degrees of Freedom:**
* df\_between (degrees of freedom between groups) = Number of groups - 1 = 3 - 1 = 2
* df\_within (degrees of freedom within groups) = Total number of observations - Number of groups = 14-3 = 11
* df\_total = N-1 = 14-1=13
**4. Calculate the Mean Square:**
* MSB (Mean Square Between Groups) = SSB / df\_between = 4.3876 / 2 = 2.1938
* MSW (Mean Square Within Groups) = SSW / df\_within = 26.602/11 = 2.418
**5. Calculate the F-statistic:**
F = MSB / MSW = 2.1938/2.418 = 0.907
**6. Determine the Critical Value:**
Using an F-distribution table or calculator with α = 0.01, df₁ = 2, and df₂ = 11, the critical F-value is approximately 7.21.
**7. Make a Decision:**
Since the calculated F-statistic (0.907) is *less than* the critical F-value (7.21), we *fail to reject* the null hypothesis.
**8. Conclusion:**
There is not sufficient evidence at the α = 0.01 level of significance to conclude that there is a significant difference in the mean sales volume per agent among the three branches. Therefore, the manager should not use this information about the sales volume in the promotion decision.
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