Exam 11: Analysis of Variance

TABLE 11-11 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 11-11, the randomized block F test is valid only if the population of diameters has the same variance for the 4 brands.

TABLE 11-10 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 Trial 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 Table 11-9, the critical value in the test for a significant interaction is_____ .

TABLE 11-10 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. 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 Table 11-10, the decision made at a 0.01 level of significance on the F test for the block effects implies that the blocking has been advantageous in reducing the experiment error.

TABLE 11-8 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 11-8, if a level of significance of 0.05 is chosen, the decision made indicates that at least 2 of the 5 locations have different mean occupancy rates.

TABLE 11-10 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 Trial 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 Table 11-10, the among-group variation or SSA is____ .

TABLE 11-10 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 Trial 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 Table 11-10, the agronomist decided to compare the 3 treatment means by using the Tukey multiple comparison procedure with an overall level of significance of 0.01. How many pair-wise comparisons can be made?

TABLE 11-6 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. Trial Barth Tornado Reiser Shaw 1 43 37 41 43 2 46 38 45 45 3 43 39 42 46 -Referring to Table 11-6, the decision made implies that all 4 means are significantly different.

TABLE 11-5 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 specialties - General Practice (GP), Internal Medicine (IM), Pediatrics (PED), and Family Physicians (FP) - and 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 USA medical school graduate. The president theorizes that foreign graduates will have higher mean charges than USA 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 USA medical school graduates) was obtained for each of the 4 specialties. The results for the ANOVA are summarized 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 Table 11-5, 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 11-10 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 11-10, the among-block variation or SSBL is____ .

TABLE 11-8 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. Caymen Pennkamp 1 28 40 21 37 22 2 33 35 21 47 19 3 41 33 27 45 25 Analysis of Variance Source SS MS F p Location 4 963.6 11.47 0.001 Error 10 210.0 Total -Referring to Table 11-8, what should be the conclusion for the Levene's test for homogeneity of variances at a 5% level of significance?

TABLE 11-10 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 11-10, using an overall level of significance of 0.01, what is the critical value of the Studentized range Q used in calculating the critical range for the Tukey multiple comparison procedure?

TABLE 11-3 A realtor wants to compare the average 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: $\quad 1.2,1.1,0.9,0.4 \quad\quad$ C: $\quad 1.0,1.5,1.1,1.3$ B: $\quad2.5,2.1,1.9,1.6 \quad\quad$ D: $\quad 0.8,1.3,1.1,0.7$ Interpret the results of the analysis summarized in the following table: Source df SS MS F PR > F Neighborhoods 3.1819 1.0606 10.76 0.001 Error 12 Total 4.3644 -Referring to Table 11-3,

TABLE 11-1 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 PR>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 11-1, in the context of this study, interpret the statement: "Subject visibility and test taker success interact."

TABLE 11-3 A realtor wants to compare the average 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 $\quad$ C: 1.0, 1.5, 1.1, 1.3 B: 2.5, 2.1, 1.9, 1.6 $\quad$ 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 PR > F Neighborhoods 3.1819 1.0606 10.76 0.001 Error 12 Total 4,3644 -Referring to Table 11-3, the p-value of the test statistic for Levene's test for homogeneity of variances 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.

TABLE 11-8 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 11-8, what is the p-value of the test statistic for Levene's test for homogeneity of variances?

TABLE 11-9 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 ? = 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 $\text { Analysis of Variance }$ 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 11-9, based on the results of the hypothesis test, it appears that there is a significant effect on the approval rating associated with the color of the logo.

TABLE 11-6 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. Trial Barth Tornado Reiser Shaw 1 43 37 41 43 2 46 38 45 45 3 43 39 42 46 -Referring to Table 11-6, the test is less sensitive to the assumption that the population of speeds has the same variance for the 4 brands if the sample sizes of the 4 brands are equal.

When the F test is used for ANOVA, the rejection region is always in the right tail.

TABLE 11-10 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. 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 Table 11-10, the randomized block F test is valid only if there is no interaction between the variety of seeds and the patches of fields.