Recitation Week 9 Autumn 2014 One Way and Two way ANOVA 1 There are a bewildering number of breakfast cereals on the market Each company produces several different products in the belief that there are distinct markets For example there is a market composed primarily of children another for diet conscious adults and another for health conscious adults Each cereal the companies produce has at least one market as it target However consumers make their own choices which may or may not match the target predicted by the cereal maker In an attempt to distinguish between consumers a survey of adults between the ages of 25 and 65 was undertaken Each was asked several questions including age income and years of education as well as which of a select set of cereals they consumed most frequently The cereal choices were 1 Sugar Smacks 2 Special K 3 Fiber One and 4 Cheerios Here we will consider whether or not there is a difference is age for the consumers of these cereals One Variable Summary Mean Variance Std Dev Skewness Median Minimum Maximum Count 1 Age 31 302 28 343 5 324 1 5194 31 000 25 000 55 000 63 2 Age 34 420 23 197 4 816 0 0311 35 000 25 000 45 000 81 3 Age 37 375 31 163 5 582 0 0021 37 000 25 000 48 000 40 4 Age 39 928 72 031 8 487 0 2550 40 000 25 000 68 000 111 a In considering a one way analysis of variance to determine if age is different by cereal type identify answer the following for this problem Factor Treatment Experimental Unit Response Variable Is this a balanced or unbalanced design Is this an experiment or an observational study b For the one way analysis of variance procedure to produce reliable information what conditions must the response variable satisfy Recitation Week 9 Autumn 2014 One Way and Two way ANOVA c Based on the problem description and the following summary information for the response variable can you say that the required conditions are met Chi Square Test Mean Std Dev Chi Square Stat P Value 1 Age 31 302 5 324 10 3973 0 0342 Chi Square Test Mean Std Dev Chi Square Stat P Value 2 Age 34 420 4 816 7 9229 0 1605 Chi Square Test Mean Std Dev Chi Square Stat P Value Chi Square Test Mean Std Dev Chi Square Stat P Value 3 Age 37 375 5 582 6 7330 0 1507 4 Age 39 928 8 487 6 5549 0 2559 Recitation Week 9 Autumn 2014 One Way and Two way ANOVA Cereal 1 Sugar Smacks 2 Special K 3 Fiber One 4 Cheerios Std Dev for Age 5 342 4 816 5 582 8 487 d Write the relevant hypotheses for the test to determine if ages of consumers are different on average by type of cereal chosen e Use the StatTools results for the one way analysis of variance to determine if at the 5 significance level the ages of consumers differ on average by type of cereal chosen ANOVA Summary Total Sample Size Grand Mean Pooled Std Dev Pooled Variance Number of Samples Confidence Level 295 36 227 6 620 43 821 4 95 00 ANOVA Sample Stats Sample Size Sample Mean Sample Std Dev Sample Variance Pooling Weight Age 1 63 31 302 5 324 28 343 0 2131 Age 2 81 34 420 4 816 23 197 0 2749 Age 3 40 37 375 5 582 31 163 0 1340 Sum of Squares 3365 986 12751 797 16117 783 Degrees of Freedom 3 291 294 Mean Squares 1121 995 43 821 OneWay ANOVA Table Between Variation Within Variation Total Variation Age 4 111 39 928 8 487 72 031 0 3780 F Ratio p Value 25 604 0 0001 Recitation Week 9 Autumn 2014 One Way and Two way ANOVA f Use the Tukey multiple comparisons to determine the nature of those differences Difference No Correction Tukey Confidence Interval Tests of Means Lower Upper Lower Upper Age 1 Age 2 3 118 6 073 8 626 2 955 5 508 2 553 5 307 8 707 10 681 5 473 7 412 4 956 0 930 3 439 6 571 0 437 3 604 0 150 5 975 9 512 11 309 6 242 7 993 5 689 0 261 2 635 5 944 0 331 3 023 0 583 Age 1 Age 3 Age 1 Age 4 Age 2 Age 3 Age 2 Age 4 Age 3 Age 4 Omitting the potential outliers ANOVA Summary Confidence Level 293 36 055 6 277 39 398 4 95 00 ANOVA Sample Stats Age 1 Cereal Preference Age 2 Cereal Preference Age 3 Cereal Preference Age 4 Cereal Preference 62 30 919 4 410 19 452 0 2111 81 34 420 4 816 23 197 0 2768 40 37 375 5 582 31 163 0 1349 110 39 673 8 087 65 396 0 3772 Sum of Degrees of Mean Squares Freedom Squares 3361 208 11385 918 14747 126 3 289 292 1120 403 39 398 Total Sample Size Grand Mean Pooled Std Dev Pooled Variance Number of Samples Sample Size Sample Mean Sample Std Dev Sample Variance Pooling Weight OneWay ANOVA Table Between Variation Within Variation Total Variation Difference F Ratio p Value 28 438 0 0001 No Correction Tukey Confidence Interval Tests of Means Lower Upper Lower Upper Age 1 Age 2 3 500 6 456 8 753 2 955 5 253 2 298 5 585 8 961 10 715 5 343 7 062 4 579 1 416 3 950 6 791 0 568 3 444 0 017 6 221 9 726 11 314 6 071 7 614 5 275 0 779 3 185 6 193 0 161 2 892 0 680 Age 1 Age 3 Age 1 Age 4 Age 2 Age 3 Age 2 Age 4 Age 3 Age 4 Recitation Week 9 Autumn 2014 One Way and Two way ANOVA 2 The vice president for engineering for a manufacturer of household washing machines wishes to determine the optimal length of time for the washing cycle as part of new product development A part of the development is to study the relationship between the detergent used four brands and the length of the washing cycle 18 20 22 or 24 minutes Thirty two standard household laundry loads having equal amounts of dirt and the same total weights are randomly assigned to the 16 detergent washing cycle combinations The results in pounds of dirt removed are shown below Detergent Brand 18 0 13 0 11 A x21 x11 B C D Cycle Time min 20 22 0 12 0 19 0 11 0 17 0 14 0 10 x12 0 16 0 17 x13 0 09 0 13 x14 x1 0 15 0 14 x22 0 15 0 14 x23 0 12 0 13 x31 0 18 0 17 x32 0 18 0 19 x33 0 16 0 16 24 0 15 0 18 x41 0 20 0 18 x42 0 19 0 21 x43 0 15 0 17 …