Exam 11: Analysis of Variance

SCENARIO 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 PR > F Neighborhoods 3.1819 1.0606 10.76 0.001 Error 12 Total 4.3644 -Referring to Scenario 11-2, the p-value of the test statistic for Levene's test for homogeneity of variances is

In a one-way ANOVA, the null hypothesis is always

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, all the assumptions needed 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 are fulfilled.

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, what should be the conclusion for the Levene's test for homogeneity of variances at a 5% level of significance?

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, based on the results of the hypothesis test, it appears that there is a significant 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, what should be the conclusion for the Levene's test for homogeneity of variances at a 5% level of significance?

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 of the test for significant differences between colors is ________.

SCENARIO 11-1 An airline wants to select a computer software package for its reservation system. Four software packages (1, 2, 3, and 4) are commercially available. The airline will choose the package that bumps as few passengers as possible during a month. An experiment is set up in which each package is used to make reservations for 5 randomly selected weeks. (A total of 20 weeks was included in the experiment.) The number of passengers bumped each week is obtained, which gives rise to the following Excel output: ANOVA Source of Variation SS df MS F P -value F crit Between Groups 212.4 3 8.304985 0.001474 3.238867 Within Groups 136.4 8.525 Total 348.8 -Referring to Scenario 11-1, at a significance level of 1%,

SCENARIO 11-6 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 -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. How Should the data be analyzed? $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$ $\quad$$\text { Starting Room }$ Wall Signs 141,119,238 224,339,139 Map 85,94,126 226,129,130

The F test statistic in a one-way ANOVA is

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 is the p-value of the test statistic for Levene's test for homogeneity of variances?

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, construct the ANOVA table from the sample data.

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 is the critical value of Levene's test for homogeneity of variances at a 5% level of significance?

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, the test is valid only if the population of crop yields is normally distributed for the 3 varieties.

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, if a level of significance of 0.05 is chosen, the decision made indicates that all 5 locations have different mean occupancy rates.

SCENARIO 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 PR > F Neighborhoods 3.1819 1.0606 10.76 0.001 Error 12 Total 4.3644 -Referring to Scenario 11-2, the among group degrees of freedom is

SCENARIO 11-7 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: $\text { ANOVA }$ Source of Variation SS df MS F P -value F crit Signs 14008.33 14008.33 0.11267 5.317645 Starting Location 12288 2.784395 0.13374 5.317645 Interaction 48 48 0.919506 5.317645 Within 35305.33 4413.167 Total 61649.67 11 -Referring to Scenario 11-7, the F test statistic for testing the main effect of types of signs 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 mean square within (MSW) of the Tukey-Kramer procedure used to determine which of the three means are different 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 total mean squares is 261.90.

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 variation or SST 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.