Exam 11: Analysis of Variance

TABLE 11-12 The marketing manager of a company producing a new cereal aimed for children wants to examine the effect of the color and shape of the box's logo on the approval rating of the cereal. He combined 4 colors and 3 shapes to produce a total of 12 designs. Each logo was presented to 2 different groups (a total of 24 groups) and the approval rating for each was recorded and is shown below. The manager analyzed these data using the α = 0.05 level of significance for all inferences. COLORS 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 Colors 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 Table 11-12, the value of the statistic used to test for significant differences between colors is ________.

TABLE 11-5 A hotel chain has identically sized resorts in 5 locations. The data that follow resulted from analyzing the hotel occupancies on randomly selected days in the 5 locations. 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 Table 11-5, the among-group variation or SSA is ________.

TABLE 11-7 A student team in a business statistics course designed an experiment to investigate whether the brand of bubblegum used affected the size of bubbles they could blow. To reduce the person-to-person variability, the students decided to use a randomized block design using themselves as blocks. Four brands of bubblegum were tested. A student chewed two pieces of a brand of gum and then blew a bubble, attempting to make it as big as possible. Another student measured the diameter of the bubble at its biggest point. The following table gives the diameters of the bubbles (in inches) for the 16 observations. Brand of Bubblegum Student A B C D Kyle 8.75 9.50 8.50 11.50 Sarah 9.50 4.00 8.50 11.00 Leigh 9.25 5.50 7.50 7.50 Isaac 9.50 8.50 7.50 7.50 -Referring to Table 11-7, is it appropriate to use the Tukey multiple comparison procedure based on the test result above?

TABLE 11-11 A physician and president of a Tampa Health Maintenance Organization (HMO) are attempting to show the benefits of managed health care to an insurance company. The physician believes that certain types of doctors are more cost-effective than others. One theory is that primary specialty is an important factor in measuring the cost-effectiveness of physicians. To investigate this, the president obtained independent random samples of 20 HMO physicians from each of 4 primary specialtiesGeneral Practice (GP), Internal Medicine (IM), Pediatrics (PED), and Family Physicians (FP)11ed1eb7_9ae3_76a5_bd82_35f3ce681344_TB1604_00and recorded the total charges per member per month for each. A second factor which the president believes influences total charges per member per month is whether the doctor is a foreign or US medical school graduate. The president theorizes that foreign graduates will have higher mean charges than US graduates. To investigate this, the president also collected data on 20 foreign medical school graduates in each of the 4 primary specialty types described above. So information on charges for 40 doctors (20 foreign and 20 US medical school graduates) was obtained for each of the 4 specialties. The results for the ANOVA are summarized in the following table. Source df 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 Table 11-11, what degrees of freedom should be used to determine the critical value of the F ratio against which to test for interaction between the two factors?

TABLE 11-8 An important factor in selecting database software is the time required for a user to learn how to use the system. To evaluate three potential brands (A, B and C) of database software, a company designed a test involving five different employees. To reduce variability due to differences among employees, each of the five employees is trained on each of the three different brands. The amount of time (in hours) needed to learn each of the three different brands is given below: Software Operator A B C 1 17 17 23 2 18 17 23 3 14 13 19 4 12 11 18 5 19 17 22 Mean 16 15 21 Below is the Excel output for the randomized block design: Source of Variation SS df MS F p -value F crit Rows 84.66667 4 21.16667 50.8 9.98-06 3.837853 Columns 103.3333 2 51.66667 124 9.54-07 4.45897 Error 3.333333 8 0.416667 Total 191.3333 14 -Referring to Table 11-8, what are the degrees of freedom of the Studentized range distribution for the Tukey multiple comparison procedure?

TABLE 11-8 An important factor in selecting database software is the time required for a user to learn how to use the system. To evaluate three potential brands (A, B and C) of database software, a company designed a test involving five different employees. To reduce variability due to differences among employees, each of the five employees is trained on each of the three different brands. The amount of time (in hours) needed to learn each of the three different brands is given below: Software Operator A B C 1 17 17 23 2 18 17 23 3 14 13 19 4 12 11 18 5 19 17 22 Mean 16 15 21 Below is the Excel output for the randomized block design: Source of Variation SS df MS F p -value F crit Rows 84.66667 4 21.16667 50.8 9.98-06 3.837853 Columns 103.3333 2 51.66667 124 9.54-07 4.45897 Error 3.333333 8 0.416667 Total 191.3333 14 -Referring to Table 11-8, there is evidence of a significant difference in the mean amount of time needed to learn Brand B and Brand C at the 5% level of significance.

TABLE 11-9 Psychologists have found that people are generally reluctant to transmit bad news to their peers. This phenomenon has been termed the "MUM effect." To investigate the cause of the MUM effect, 40 undergraduates at Duke University participated in an experiment. Each subject was asked to administer an IQ test to another student and then provide the test taker with his or her percentile score. Unknown to the subject, the test taker was a bogus student who was working with the researchers. The experimenters manipulated two factors: subject visibility and success of test taker, each at two levels. Subject visibility was either visible or not visible to the test taker. Success of the test taker was either top 20% or bottom 20%. Ten subjects were randomly assigned to each of the 2 x 2 = 4 experimental conditions, then the time (in seconds) between the end of the test and the delivery of the percentile score from the subject to the test taker was measured. (This variable is called the latency to feedback.) The data were subjected to appropriate analyses with the following results. Source df SS MS F > Subject visibility 1 1380.24 1380.24 4.26 0.043 Test taker success 1 1325.16 1325.16 4.09 0.050 Interaction 1 3385.80 3385.80 10.45 0.002 Error 36 11,664.00 324.00 Total 39 17,755.20 -Referring to Table 11-9, in the context of this study, interpret the statement: "Subject visibility and test taker success interact."

TABLE 11-3 As part of an evaluation program, a sporting goods retailer wanted to compare the downhill coasting speeds of 4 brands of bicycles. She took 3 of each brand and determined their maximum downhill speeds. The results are presented in miles per hour in the table below. 1 43 37 41 43 2 46 38 45 45 3 43 39 42 46 -Referring to Table 11-3, the test is less sensitive to the assumption that the population of speeds has the same variance for the 4 brands because the sample sizes of the 4 brands are equal.

TABLE 11-7 A student team in a business statistics course designed an experiment to investigate whether the brand of bubblegum used affected the size of bubbles they could blow. To reduce the person-to-person variability, the students decided to use a randomized block design using themselves as blocks. Four brands of bubblegum were tested. A student chewed two pieces of a brand of gum and then blew a bubble, attempting to make it as big as possible. Another student measured the diameter of the bubble at its biggest point. The following table gives the diameters of the bubbles (in inches) for the 16 observations. Brand of Bubblegum Student A B C D Kyle 8.75 9.50 8.50 11.50 Sarah 9.50 4.00 8.50 11.00 Leigh 9.25 5.50 7.50 7.50 Isaac 9.50 8.50 7.50 7.50 -Referring to Table 11-7, what is the p-value of the test statistic for the randomized block F test for the difference in the means?

Which of the following components in an ANOVA table are not additive?

TABLE 11-5 A hotel chain has identically sized resorts in 5 locations. The data that follow resulted from analyzing the hotel occupancies on randomly selected days in the 5 locations. 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 -A completely randomized design with 4 groups would have 6 possible pairwise comparisons.

TABLE 11-3 As part of an evaluation program, a sporting goods retailer wanted to compare the downhill coasting speeds of 4 brands of bicycles. She took 3 of each brand and determined their maximum downhill speeds. The results are presented in miles per hour in the table below. 1 43 37 41 43 2 46 38 45 45 3 43 39 42 46 -Referring to Table 11-3, the null hypothesis should be rejected at a 5% level of significance.

TABLE 11-3 As part of an evaluation program, a sporting goods retailer wanted to compare the downhill coasting speeds of 4 brands of bicycles. She took 3 of each brand and determined their maximum downhill speeds. The results are presented in miles per hour in the table below. 1 43 37 41 43 2 46 38 45 45 3 43 39 42 46 -Referring to Table 11-3, the test is valid only if the population of speeds is normally distributed.

TABLE 11-6 An agronomist wants to compare the crop yield of 3 varieties of chickpea seeds. She plants all 3 varieties of the seeds on each of 5 different patches of fields. She then measures the crop yield in bushels per acre. Treating this as a randomized block design, the results are presented in the table that follows. 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 Table 11-6, based on the Tukey multiple comparison procedure with an overall level of significance of 0.01, the agronomist would decide that there is a significant difference between the crop yield of Walsh and Trevor seeds.

TABLE 11-3 As part of an evaluation program, a sporting goods retailer wanted to compare the downhill coasting speeds of 4 brands of bicycles. She took 3 of each brand and determined their maximum downhill speeds. The results are presented in miles per hour in the table below. 1 43 37 41 43 2 46 38 45 45 3 43 39 42 46 -Referring to Table 11-3, what is the critical value of Levene's test for homogeneity of variances at a 5% level of significance?

TABLE 11-4 An agronomist wants to compare the crop yield of 3 varieties of chickpea seeds. She plants 15 fields, 5 with each variety. She then measures the crop yield in bushels per acre. Treating this as a completely randomized design, the results are presented in the table that follows. 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 Table 11-4, the within-group variation or SSW is ________.

TABLE 11-6 An agronomist wants to compare the crop yield of 3 varieties of chickpea seeds. She plants all 3 varieties of the seeds on each of 5 different patches of fields. She then measures the crop yield in bushels per acre. Treating this as a randomized block design, the results are presented in the table that follows. 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 Table 11-6, the among-block variation or SSBL is ________.

TABLE 11-8 An important factor in selecting database software is the time required for a user to learn how to use the system. To evaluate three potential brands (A, B and C) of database software, a company designed a test involving five different employees. To reduce variability due to differences among employees, each of the five employees is trained on each of the three different brands. The amount of time (in hours) needed to learn each of the three different brands is given below: Software Operator A B C 1 17 17 23 2 18 17 23 3 14 13 19 4 12 11 18 5 19 17 22 Mean 16 15 21 Below is the Excel output for the randomized block design: Source of Variation SS df MS F p -value F crit Rows 84.66667 4 21.16667 50.8 9.98-06 3.837853 Columns 103.3333 2 51.66667 124 9.54-07 4.45897 Error 3.333333 8 0.416667 Total 191.3333 14 -Referring to Table 11-8, what is the upper-tail critical value of the Studentized range distribution for the Tukey multiple comparison procedure at the 5% level of significance?

TABLE 11-2 A realtor wants to compare the mean sales-to-appraisal ratios of residential properties sold in four neighborhoods (A, B, C, and D). Four properties are randomly selected from each neighborhood 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 summarized in the following table: Source df SS MS F > Neighborhoods 3.1819 1.0606 10.76 0.001 Error 12 Total 4.3644 -Referring to Table 11-2, what should be the conclusion for the Levene's test for homogeneity of variances at a 5% level of significance?

TABLE 11-10 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 -Referring to Table 11-10, the F test statistic for testing the main effect of types of signs is