Exam 11: Analysis of Variance

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, the null hypothesis for the F test for the block effects should be rejected at a 0.05 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, in testing the null hypothesis that the mean downhill coasting speeds of the 4 brands of bicycles are equal, the value of the test statistic is ________.

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, the value of the test statistic for Levene's test for homogeneity of variances is

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 assumption(s)need(s)to be made in order to conduct the test for differences between the mean charges of foreign and US medical school graduates?

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 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 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, the relative efficiency means that 15.23 times as many observations in each brand would be needed in a one-way ANOVA design as compared to the randomized block design in order to obtain the same precision for comparison of the different means.

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 null hypothesis should be rejected at 0.005 level of significance.

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, the randomized block F test is valid only if there is no interaction between the diameters of the 4 brands of bubble gums and the 4 ability of the students.

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 critical value of the test for significant differences between colors 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, the amount of total variation or SST 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, the decision made at a 0.05 level of significance on the randomized block F test for the difference in means implies that all 4 means are significantly different.

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 null hypothesis for the randomized block F test for the difference in the means 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 critical value of the randomized block F test for the difference in the means at a level of significance of 0.05?