Exam 11: Analysis of Variance

SCENARIO 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 Scenario 11-4, based on the Tukey-Kramer 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 Smith and Trevor seeds.

SCENARIO 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 Scenario 11-3, what are the numerator and denominator degrees of freedom for Levene's test for homogeneity of variances respectively?

SCENARIO 11-10-A You are the quality control manager of a water bottles company. One of the biggest complaints in the past years has been the breakage and, hence, the concern on the durability of the connector between the lid and the bottle which many users use as a handle for the bottles. You have started looking for alternative replacement for the existing connector design. Three different vendors have proposed different designs and provided data they have collected on their designs that have been subjected to a durability test. The data on the number of times the connectors designed by them have been used to lift the bottles before they break from random samples of 30 bottles are shown below and also contained in the file Scenario11-10-DataA.XLSX. Vendor A 1498 1667 1530 1458 1315 1458 1411 1348 1583 1565 1495 1533 1473 1654 1499 1349 1486 1739 1820 1543 1440 1652 1610 1660 1563 1507 1450 1501 1515 1501 Vendor B 2027 2007 1977 1787 2046 1945 1898 2130 2096 1876 2014 1815 1982 2004 1918 1904 2061 1986 1952 2061 2038 2227 2030 2079 1884 2045 2009 1970 1946 2011 Vendor C 2784 3151 3126 2875 2881 3000 2872 3097 3020 3077 3082 2931 2922 2877 2905 3079 2916 2917 3110 3121 3084 3065 2974 2900 2860 2989 2978 2881 3095 2893 Assume that and the number of times the connectors have been used to lift the bottles before they break follows a normal distribution. You want to test to see if there is enough evidence of a difference in mean number of times the connectors have been used to lift the bottles before they break across the three vendors. -Referring to Scenario 11-10-A, the among-group variation or SSA is _________ for the test to determine whether there is enough evidence of a difference in mean number of times the connectors have been used to lift the bottles before they break across the three vendors.

SCENARIO 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 α = 0.05 level of significance for all inferences. $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$$\text {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 Scenario 11-9, the critical value in the test for significant differences between shapes is ________.

SCENARIO 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 α = 0.05 level of significance for all inferences. $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$$\text {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 Scenario 11-9, the value of the statistic used to test for significant differences between shapes is ________.

In a two-way ANOVA, the degrees of freedom for the "error" term is

The MSE must always be positive.

SCENARIO 11-5 A hotel chain has identically small sized resorts in 5 locations in different small islands. The data that follow resulted from analyzing the hotel occupancies on randomly selected days in the 5 locations. 1 28 40 21 37 2 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 Scenario 11-5, what is the p-value of the test statistic for Levene's test for homogeneity of variances?

SCENARIO 11-10-B You are the quality control manager of a water bottles company. One of the biggest complaints in the past years has been the breakage and, hence, the concern on the durability of the connector between the lid and the bottle which many users use as a handle for the bottles. You have started looking for alternative replacement for the existing connector design. Three different vendors have proposed different designs and provided data they have collected on their designs that have been subjected to a durability test. The data on the number of times the connectors designed by them have been used to lift the bottles before they break from random samples of 30 bottles are shown below and also contained in the file Scenario11-10-DataB.XLSX. Vendor A 1460 1519 1364 1521 1628 1265 1552 1496 1411 1614 1508 1587 1484 1554 1459 1529 1515 1446 1517 1511 1563 1548 1536 1478 1665 1656 1544 1361 1573 1429 Vendor B 2941 2975 3231 2890 3001 2930 2924 2916 3048 3066 3012 3108 2982 3126 3105 2879 2944 3066 2984 3057 3217 3058 2987 2977 3178 3252 3024 3156 3029 3103 Vendor C 2020 1928 2135 1777 2005 2061 2029 1860 1967 2099 2031 2188 2137 1945 1952 1905 2031 1853 1974 2071 1966 2084 1998 2179 1988 2002 2178 1995 2076 1873 Assume that and the number of times the connectors have been used to lift the bottles before they break follows a normal distribution. You want to test to see if there is enough evidence of a difference in mean number of times the connectors have been used to lift the bottles before they break across the three vendors. -Referring to Scenario 11-10-B, the mean square within (MSW) of the Tukey-Kramer procedure used to determine which of the three means are different is ________.

SCENARIO 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 α = 0.05 level of significance for all inferences. $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$$\text {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 Scenario 11-9, the mean square for the interaction of color and shape is ________.

SCENARIO 11-5 A hotel chain has identically small sized resorts in 5 locations in different small islands. The data that follow resulted from analyzing the hotel occupancies on randomly selected days in the 5 locations. 1 28 40 21 37 2 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 Scenario 11-5, the among-group variation or SSA is _________.

If you are comparing the mean sales among 3 different brands you are dealing with a three-way ANOVA design.

SCENARIO 11-8 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 Scenario 11-8, interpret the test for interaction.

SCENARIO 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 Scenario 11-3, the test is valid only if the population of speeds is normally distributed.

SCENARIO 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 Scenario 11-3, the test is robust to the violation of the assumption that the population of speeds is normally distributed.

SCENARIO 11-10-B You are the quality control manager of a water bottles company. One of the biggest complaints in the past years has been the breakage and, hence, the concern on the durability of the connector between the lid and the bottle which many users use as a handle for the bottles. You have started looking for alternative replacement for the existing connector design. Three different vendors have proposed different designs and provided data they have collected on their designs that have been subjected to a durability test. The data on the number of times the connectors designed by them have been used to lift the bottles before they break from random samples of 30 bottles are shown below and also contained in the file Scenario11-10-DataB.XLSX. Vendor A 1460 1519 1364 1521 1628 1265 1552 1496 1411 1614 1508 1587 1484 1554 1459 1529 1515 1446 1517 1511 1563 1548 1536 1478 1665 1656 1544 1361 1573 1429 Vendor B 2941 2975 3231 2890 3001 2930 2924 2916 3048 3066 3012 3108 2982 3126 3105 2879 2944 3066 2984 3057 3217 3058 2987 2977 3178 3252 3024 3156 3029 3103 Vendor C 2020 1928 2135 1777 2005 2061 2029 1860 1967 2099 2031 2188 2137 1945 1952 1905 2031 1853 1974 2071 1966 2084 1998 2179 1988 2002 2178 1995 2076 1873 Assume that and the number of times the connectors have been used to lift the bottles before they break follows a normal distribution. You want to test to see if there is enough evidence of a difference in mean number of times the connectors have been used to lift the bottles before they break across the three vendors. -Referring to Scenario 11-10-B, the numerator and denominator degrees of freedom of the test ratio are ________ and ________, respectively, for the test to determine whether there is enough evidence of a difference in mean number of times the connectors have been used to lift the bottles before they break across the three vendors.

SCENARIO 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 Scenario 11-4, state the null hypothesis that can be tested.

In a two-way ANOVA, the degrees of freedom for the interaction term is

SCENARIO 11-10-A You are the quality control manager of a water bottles company. One of the biggest complaints in the past years has been the breakage and, hence, the concern on the durability of the connector between the lid and the bottle which many users use as a handle for the bottles. You have started looking for alternative replacement for the existing connector design. Three different vendors have proposed different designs and provided data they have collected on their designs that have been subjected to a durability test. The data on the number of times the connectors designed by them have been used to lift the bottles before they break from random samples of 30 bottles are shown below and also contained in the file Scenario11-10-DataA.XLSX. Vendor A 1498 1667 1530 1458 1315 1458 1411 1348 1583 1565 1495 1533 1473 1654 1499 1349 1486 1739 1820 1543 1440 1652 1610 1660 1563 1507 1450 1501 1515 1501 Vendor B 2027 2007 1977 1787 2046 1945 1898 2130 2096 1876 2014 1815 1982 2004 1918 1904 2061 1986 1952 2061 2038 2227 2030 2079 1884 2045 2009 1970 1946 2011 Vendor C 2784 3151 3126 2875 2881 3000 2872 3097 3020 3077 3082 2931 2922 2877 2905 3079 2916 2917 3110 3121 3084 3065 2974 2900 2860 2989 2978 2881 3095 2893 Assume that and the number of times the connectors have been used to lift the bottles before they break follows a normal distribution. You want to test to see if there is enough evidence of a difference in mean number of times the connectors have been used to lift the bottles before they break across the three vendors. -Referring to Scenario 11-10-A, the total mean squares is ________ for the test to determine whether there is enough evidence of a difference in mean number of times the connectors have been used to lift the bottles before they break across the three vendors.

SCENARIO 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 α = 0.05 level of significance for all inferences. $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$$\text {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 Scenario 11-9, the mean square for the factor color is ________.