12.7 Assignment 12

Purposes

The following questions have two parts. The first part assesses your knowledge of identifying cases where one-way ANOVA should be used, explaining the main idea of one-way ANOVA, and conducting a one-way ANOVA F test based on the computer outputs. The second part assesses your skills in using R Commander to conduct a one-way ANOVA F test.

Resources

M12_Rent_ANOVA_Q5_FourColumnS.xlsx

M12_Salary_ANOVA_Q7.xlsx

M12_Rent_ANOVA_Q6.xlsx

Instructions

Part A

Complete the following:

1. What does ANOVA stand for? (2 marks)
2. When shall we use a one-way ANOVA F test? (2 marks)
3. Suppose that a one-way ANOVA is being performed to compare the means of three populations and that the sample sizes are 10, 12, and 15. Determine the degrees of freedom for the F statistic. (2 marks)
4. We stated earlier that a one-way ANOVA test is always right-tailed because the null hypothesis is rejected only when the test statistic, F, is too large. Why is the null hypothesis rejected only when F is too large? (3 marks)
5. Fill in the missing entries in the following ANOVA table. (4 marks)
   

   Source	df	SS	MS = [latex]\frac{SS}{df}[/latex]	F-statistic
   Treatment	2	?	21.652	[latex]F_{o} = ?[/latex]
   Error	?	84.400
   Total	14	?

6. The data on monthly rents, in dollars, for independent random samples of newly completed apartments in the four U.S. regions are presented in the following table. (See the spreadsheet M12_Rent_ANOVA_Q5_FourColumnS.xlsx)
   

   Northeast	Midwest	South	West
   1005	870	891	1025
   898	748	630	1012
   948	699	861	1090
   1181	814	1036	926
   1244	721		1269
   	606

   Given the ANOVA table, test at a 5% significance level whether a difference exists in the mean rent of newly completed apartments in the four U.S. regions. (8 marks)

   Source	df	SS	MS = [latex]\frac{SS}{df}[/latex]	F-statistic	P-value
   Region	3	400513	133504	[latex]F_{o} = 7.541[/latex]	0.0023
   Error	16	283265	17704
   Total	19	683778

7. The following table gives the salaries (in thousand dollars) for computer science (CS) majors obtaining a bachelor’s degree, a master’s degree, or a Ph.D. (See the spreadsheet M12_Salary_ANOVA_Q7.xlsx)
   

   Salary	Degree	Salary	Degree	Salary	Degree
   50.8	Bachelor’s	65.8	Master’s	73.3	Ph.D
   59.4	Bachelor’s	57.5	Master’s	65.7	Ph.D
   55.9	Bachelor’s	66.9	Master’s	71.7	Ph.D
   45.1	Bachelor’s	62.8	Master’s	72.5	Ph.D
   54.1	Bachelor’s	68.5	Master’s	73.0	Ph.D
   50.7	Bachelor’s	69.3	Master’s	67.2	Ph.D
   46.8	Bachelor’s	61.5	Master’s	67.5	Ph.D

   1. Fill in missing entries of the following ANOVA table. (4 marks)
      

      Source	df	SS	MS = [latex]\frac{SS}{df}[/latex]	F-statistic	P-value
      Degree	?	?	616.9	[latex]F_{o} = 34.62[/latex]	< 0.0001
      Error	?	?	17.8
      Total	20	1554.5

   2. Test at the 1% significance level whether a difference exists in mean salary for computer science (CS) majors obtaining a bachelor’s degree, a master’s degree or a Ph.D. (8 marks)

Part B

Finish the following questions using R and R commander.

1. Refer to Question 6 in Part A. The data are provided in the file M12_Rent_ANOVA_Q6.xlsx. Import the data into R commander. Regenerate the ANOVA table in Question 6 in Part A. Make sure you copy and paste the computer output. (3 marks)
2. Refer to Question 7 in Part A. The data are provided in the file M12_Salary_ANOVA_Q7.xlsx. Import the data into R commander. Obtain the ANOVA table and compare it with Question 7 in Part A. Make sure you copy and paste the computer output. (4 marks)