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. (1) (2) (3) (4) One Variable Summary Age Age Age Age Mean 31.302 34.420 37.375 39.928 Variance 28.343 23.197 31.163 72.031 Std. Dev. 5.324 4.816 5.582 8.487 Skewness 1.5194 0.0311 0.0021 0.2550 Median 31.000 35.000 37.000 40.000 Minimum 25.000 25.000 25.000 25.000 Maximum 55.000 45.000 48.000 68.000 Count 63 81 40 111 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? (1) Chi-Square Test Age Mean 31.302 Std Dev 5.324 Chi-Square Stat. 10.3973 P-Value 0.0342 (2) Chi-Square Test Age Mean 34.420 Std Dev 4.816 Chi-Square Stat. 7.9229 P-Value 0.1605 (3) Chi-Square Test Age Mean 37.375 Std Dev 5.582 Chi-Square Stat. 6.7330 P-Value 0.1507 (4) Chi-Square Test Age Mean 39.928 Std Dev 8.487 Chi-Square Stat. 6.5549 P-Value 0.2559Recitation – Week 9 – Autumn 2014 One-Way and Two-way ANOVA 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 295 Grand Mean 36.227 Pooled Std Dev 6.620 Pooled Variance 43.821 Number of Samples 4 Confidence Level 95.00% ANOVA Sample Stats Age (1) Age (2) Age (3) Age (4) Sample Size 63 81 40 111 Sample Mean 31.302 34.420 37.375 39.928 Sample Std Dev 5.324 4.816 5.582 8.487 Sample Variance 28.343 23.197 31.163 72.031 Pooling Weight 0.2131 0.2749 0.1340 0.3780 Sum of Degrees of Mean F-Ratio p-Value OneWay ANOVA Table Squares Freedom Squares Between Variation 3365.986 3 1121.995 25.604 < 0.0001 Within Variation 12751.797 291 43.821 Total Variation 16117.783 294 Cereal Std Dev for Age 1 = Sugar Smacks 5.342 2 = Special K 4.816 3 = Fiber One 5.582 4 = Cheerios 8.487Recitation – 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 -5.307 -0.930 -5.975 -0.261 Age (1)-Age (3) -6.073 -8.707 -3.439 -9.512 -2.635 Age (1)-Age (4) -8.626 -10.681 -6.571 -11.309 -5.944 Age (2)-Age (3) -2.955 -5.473 -0.437 -6.242 0.331 Age (2)-Age (4) -5.508 -7.412 -3.604 -7.993 -3.023 Age (3)-Age (4) -2.553 -4.956 -0.150 -5.689 0.583 Omitting the potential outliers: ANOVA Summary Total Sample Size 293 Grand Mean 36.055 Pooled Std Dev 6.277 Pooled Variance 39.398 Number of Samples 4 Confidence Level 95.00% Age (1) Age (2) Age (3) Age (4) ANOVA Sample Stats Cereal Preference Cereal Preference Cereal Preference Cereal Preference Sample Size 62 81 40 110 Sample Mean 30.919 34.420 37.375 39.673 Sample Std Dev 4.410 4.816 5.582 8.087 Sample Variance 19.452 23.197 31.163 65.396 Pooling Weight 0.2111 0.2768 0.1349 0.3772 Sum of Degrees of Mean F-Ratio p-Value OneWay ANOVA Table Squares Freedom Squares Between Variation 3361.208 3 1120.403 28.438 < 0.0001 Within Variation 11385.918 289 39.398 Total Variation 14747.126 292 Difference No Correction Tukey Confidence Interval Tests of Means Lower Upper Lower Upper Age (1)-Age (2) -3.500 -5.585 -1.416 -6.221 -0.779 Age (1)-Age (3) -6.456 -8.961 -3.950 -9.726 -3.185 Age (1)-Age (4) -8.753 -10.715 -6.791 -11.314 -6.193 Age (2)-Age (3) -2.955 -5.343 -0.568 -6.071 0.161 Age (2)-Age (4) -5.253 -7.062 -3.444 -7.614 -2.892 Age (3)-Age (4) -2.298 -4.579 -0.017 -5.275 0.680Recitation – 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. Cycle Time (min) Detergent Brand 18 20 22 24 A 0.13 0.12 0.19 0.15 1.x= 0.11 0.11 0.17 0.18 11x 21x 31x 41x B 0.14 0.15 0.18 0.20 2.x= 0.10 0.14 0.17 0.18 12x= 22x 32x 42x C 0.16 0.15 0.18 0.19 3.x= 0.17 0.14 0.19 0.21 13x= 23x 33x 43x D 0.09 0.12 0.16 0.15 4.x= 0.13 0.13 0.16 0.17 14x= 24x 34x 44x .1x= .2x= .3x= .4x= x= Is this an Experiment or an Observational Study? Is this a Balanced or Unbalanced design? Identify the Response (“dependent”) Variable. Identify Factor A (“independent” variable). Identify Factor B (“independent” variable). Sum