Exam 11: Analysis of Variance

Instruction 11-3 A realtor wants to compare the average sales-to-appraisal ratios of residential properties sold in four neighbourhoods (A,B,C,and D).Four properties are randomly selected from each neighbourhood and the ratios recorded for each,as shown below. A: 1.2,1.1,0.9,0.4 C: 1.0,1.5,1.1,1.3 B: 2.5,2.1,1.9,1.6 D: 0.8,1.3,1.1,0.7 Interpret the results of the analysis summarised in the following table: Source df SS MS F PR > F Neighbourhoods 3.1819 1.0606 10.76 0.001 Error 12 total 4.3644 -Referring to Instruction 11-3,the value of the test statistic for Levene's test for homogeneity of variances is

Instruction 11-8 An agronomist wants to compare the crop yield of three varieties of chickpea seeds.She plants all three varieties of the seeds on each of five different patches of fields.She then measures the crop yield in bushels per acre.Treating this as a randomised block design,the results are presented in the table that follows. Fields Smith Walsh Trevor 1 11.1 19.0 14.6 2 13.5 18.0 15.7 3 15.3 19.8 16.8 4 14.6 19.6 16.7 5 9.8 16.6 15.2 -Referring to Instruction 11-8,what is the value of the F test statistic for testing the block effects?

Instruction 11-6 As part of an evaluation program,a sporting goods retailer wanted to compare the downhill coasting speeds of four brands of bicycles.She recorded three trials of each brand and determined their maximum downhill speeds.The results are presented in kilometres per hour in the table below. Trial 1 43 37 41 43 2 46 38 45 45 3 43 39 42 46 -Referring to Instruction 11-6,in testing the null hypothesis that the average downhill coasting speeds of the four brands of bicycles are equal,the value of the test statistic is ________.

Instruction 11-7 A hotel chain has identically sized resorts in five locations.The following data resulted from analysing the hotel guests on randomly selected days in the five locations. ROW Caymen Pennkapmp California Mayaguez Maui 1 28 40 21 37 22 2 33 35 21 47 19 3 41 33 27 45 25 Analysis of Variance Source df SS MS F p Location 4 963.6 11.47 0.001 Error 10 210.0 Total -Referring to Instruction 11-7,if a level of significance of 0.05 is chosen,the decision made indicates that at least two of the five locations have different mean occupancy rates.

Instruction 11-8 An agronomist wants to compare the crop yield of three varieties of chickpea seeds.She plants all three varieties of the seeds on each of five different patches of fields.She then measures the crop yield in bushels per acre.Treating this as a randomised block design,the results are presented in the table that follows. Fields Smith Walsh Trevor 1 11.1 19.0 14.6 2 13.5 18.0 15.7 3 15.3 19.8 16.8 4 14.6 19.6 16.7 5 9.8 16.6 15.2 -Referring to Instruction 11-8,the among-block variation or SSBL is ________.

In a two-factor ANOVA analysis,the sum of squares due to both factors,the interaction sum of squares and the within sum of squares must add up to the total sum of squares.

Instruction 11-10 The marketing manager of a company producing a new cereal aimed for children wants to examine the effect of the colour and shape of the box's logo on the approval rating of the cereal.He combined four colours and three shapes to produce a total of 12 designs.Each logo was presented to two different groups (a total of 24 groups)and the approval rating for each was recorded and is shown below.The manager analysed these data using the α = 0.05 level of significance for all inferences. COLOURS SHAPES Red Green Blue Yellow Circle 54 67 36 45 44 61 44 41 Square 34 56 36 21 36 58 30 25 Diamond 46 60 34 31 48 60 38 33 Analysis of Variance Source df SS MS F p Colours 3 2711.17 903.72 72.30 0.000 Shapes 2 579.00 289.50 23.16 0.000 Interaction 6 150.33 25.06 2.00 0.144 Error 12 150.00 12.50 Total 23 3590.50 -Referring to Instruction 11-10,the critical value in the test for significant differences between shapes is ________.

Instruction 11-10 The marketing manager of a company producing a new cereal aimed for children wants to examine the effect of the colour and shape of the box's logo on the approval rating of the cereal.He combined four colours and three shapes to produce a total of 12 designs.Each logo was presented to two different groups (a total of 24 groups)and the approval rating for each was recorded and is shown below.The manager analysed these data using the α = 0.05 level of significance for all inferences. COLOURS SHAPES Red Green Blue Yellow Circle 54 67 36 45 44 61 44 41 Square 34 56 36 21 36 58 30 25 Diamond 46 60 34 31 48 60 38 33 Analysis of Variance Source df SS MS F p Colours 3 2711.17 903.72 72.30 0.000 Shapes 2 579.00 289.50 23.16 0.000 Interaction 6 150.33 25.06 2.00 0.144 Error 12 150.00 12.50 Total 23 3590.50 -Referring to Instruction 11-10,the mean square for the factor colour is ________.

Instruction 11-7 A hotel chain has identically sized resorts in five locations.The following data resulted from analysing the hotel guests on randomly selected days in the five locations. ROW Caymen Pennkapmp California Mayaguez Maui 1 28 40 21 37 22 2 33 35 21 47 19 3 41 33 27 45 25 Analysis of Variance Source df SS MS F p Location 4 963.6 11.47 0.001 Error 10 210.0 Total -Referring to Instruction 11-7,what is the p-value of the test statistic for Levene's test for homogeneity of variances?

Instruction 11-8 An agronomist wants to compare the crop yield of three varieties of chickpea seeds.She plants all three varieties of the seeds on each of five different patches of fields.She then measures the crop yield in bushels per acre.Treating this as a randomised block design,the results are presented in the table that follows. Fields Smith Walsh Trevor 1 11.1 19.0 14.6 2 13.5 18.0 15.7 3 15.3 19.8 16.8 4 14.6 19.6 16.7 5 9.8 16.6 15.2 -Referring to Instruction 11-8,what are the degrees of freedom of the F test statistic for testing the block effects?

Instruction 11-7 A hotel chain has identically sized resorts in five locations.The following data resulted from analysing the hotel guests on randomly selected days in the five locations. ROW Caymen Pennkapmp California Mayaguez Maui 1 28 40 21 37 22 2 33 35 21 47 19 3 41 33 27 45 25 Analysis of Variance Source df SS MS F p Location 4 963.6 11.47 0.001 Error 10 210.0 Total -Referring to Instruction 11-7,the numerator and denominator degrees of freedom of the test ratio are ________ and ________,respectively.

A completely randomised design

Instruction 11-5 The CEO of a large Australian health fund is attempting to show the benefits of managed health care to an insurance underwriting company.The CEO believes that certain types of doctors are more cost-effective than others.One theory states that Primary Specialty is an important factor in measuring the cost-effectiveness of physicians.To investigate this,the CEO obtained independent random samples of 20 physicians from each of four primary specialties - General Practice (GP),Internal Medicine (IM),Paediatrics (PED),and Family Physicians (FP)- and recorded the total charges per member per month for each.A second factor which the CEO believes influences total charges per member per month is whether the doctor is a foreign or Australian medical school graduate.The CEO theorises that foreign graduates will have higher mean charges than Australian graduates.To investigate this,the CEO also collected data on 20 foreign medical school graduates in each of the four primary specialty types described above.So information on charges for 40 doctors (20 foreign and 20 Australian medical school graduates)was obtained for each of the four specialties.The results for the ANOVA are summarised in the following table. Source df SS MS F PR > F Specialty 3 22,855 7,618 60.94 0.0001 Med school 1 105 105 0.84 0.6744 Interaction 3 890 297 2.38 0.1348 Error 152 18,950 Total 159 42,800 -Referring to Instruction 11-5,is there evidence of a difference between the mean charges of foreign and Australian medical school graduates?

Instruction 11-6 As part of an evaluation program,a sporting goods retailer wanted to compare the downhill coasting speeds of four brands of bicycles.She recorded three trials of each brand and determined their maximum downhill speeds.The results are presented in kilometres per hour in the table below. Trial 1 43 37 41 43 2 46 38 45 45 3 43 39 42 46 -Referring to Instruction 11-6,the decision made implies that all four means are significantly different.

The F test statistic in a one-way ANOVA is

Instruction 11-4 A campus researcher wanted to investigate the factors that affect visitor travel time in a complex,multilevel building on campus.Specifically,he wanted to determine whether different building signs (building maps versus wall signage)affect the total amount of time visitors require to reach their destination and whether that time depends on whether the starting location is inside or outside the building.Three subjects were assigned to each of the combinations of signs and starting locations,and travel time in seconds from beginning to destination was recorded.An Excel output of the appropriate analysis is given below: ANOVA Source of Variation SS df MS F P-value Fcrit Signs 14008.33 14008.33 0.11267 5.317645 Starting Location 12288 2.784395 0.13374 5.317645 -Referring to Instruction 11-4,at 1% level of significance,

Instruction 11-7 A hotel chain has identically sized resorts in five locations.The following data resulted from analysing the hotel guests on randomly selected days in the five locations. ROW Caymen Pennkapmp California Mayaguez Maui 1 28 40 21 37 22 2 33 35 21 47 19 3 41 33 27 45 25 Analysis of Variance Source df SS MS F p Location 4 963.6 11.47 0.001 Error 10 210.0 Total -Referring to Instruction 11-7,the total variation or SST is ________.

Instruction 11-5 The CEO of a large Australian health fund is attempting to show the benefits of managed health care to an insurance underwriting company.The CEO believes that certain types of doctors are more cost-effective than others.One theory states that Primary Specialty is an important factor in measuring the cost-effectiveness of physicians.To investigate this,the CEO obtained independent random samples of 20 physicians from each of four primary specialties - General Practice (GP),Internal Medicine (IM),Paediatrics (PED),and Family Physicians (FP)- and recorded the total charges per member per month for each.A second factor which the CEO believes influences total charges per member per month is whether the doctor is a foreign or Australian medical school graduate.The CEO theorises that foreign graduates will have higher mean charges than Australian graduates.To investigate this,the CEO also collected data on 20 foreign medical school graduates in each of the four primary specialty types described above.So information on charges for 40 doctors (20 foreign and 20 Australian medical school graduates)was obtained for each of the four specialties.The results for the ANOVA are summarised in the following table. Source df SS MS F PR > F Specialty 3 22,855 7,618 60.94 0.0001 Med school 1 105 105 0.84 0.6744 Interaction 3 890 297 2.38 0.1348 Error 152 18,950 Total 159 42,800 -Referring to Instruction 11-5,what assumption(s)need(s)to be made in order to conduct the test for differences between the mean charges of foreign and Australian medical school graduates?

Instruction 11-10 The marketing manager of a company producing a new cereal aimed for children wants to examine the effect of the colour and shape of the box's logo on the approval rating of the cereal.He combined four colours and three shapes to produce a total of 12 designs.Each logo was presented to two different groups (a total of 24 groups)and the approval rating for each was recorded and is shown below.The manager analysed these data using the α = 0.05 level of significance for all inferences. COLOURS SHAPES Red Green Blue Yellow Circle 54 67 36 45 44 61 44 41 Square 34 56 36 21 36 58 30 25 Diamond 46 60 34 31 48 60 38 33 Analysis of Variance Source df SS MS F p Colours 3 2711.17 903.72 72.30 0.000 Shapes 2 579.00 289.50 23.16 0.000 Interaction 6 150.33 25.06 2.00 0.144 Error 12 150.00 12.50 Total 23 3590.50 -Referring to Instruction 11-10,based on the results of the hypothesis test,it appears that there is a significant interaction.

Instruction 11-10 The marketing manager of a company producing a new cereal aimed for children wants to examine the effect of the colour and shape of the box's logo on the approval rating of the cereal.He combined four colours and three shapes to produce a total of 12 designs.Each logo was presented to two different groups (a total of 24 groups)and the approval rating for each was recorded and is shown below.The manager analysed these data using the α = 0.05 level of significance for all inferences. COLOURS SHAPES Red Green Blue Yellow Circle 54 67 36 45 44 61 44 41 Square 34 56 36 21 36 58 30 25 Diamond 46 60 34 31 48 60 38 33 Analysis of Variance Source df SS MS F p Colours 3 2711.17 903.72 72.30 0.000 Shapes 2 579.00 289.50 23.16 0.000 Interaction 6 150.33 25.06 2.00 0.144 Error 12 150.00 12.50 Total 23 3590.50 -Referring to Instruction 11-10,the critical value in the test for a significant interaction is ________.