Factorial ANOVA Problems Q.1. In a 2-Factor ANOVA, measuring the effects of 2 factors (A and B) on a response (y), there are 3 levels each for factors A and B, and 4 replications per treatment combination. Give the values of of the F-statistic for the AB interaction for which we will conclude the effects of Factor A levels depend on Factor B levels and vice versa ( =0.05): Q.2. The following partial ANOVA table was obtained from a 2-way ANOVA where three advertisements were being compared among men and women. A total of 30 males and 30 females were sampled, and 10 of each were exposed to ads 1, 2, and 3, respectively. A measure of attitude toward the brand was obtained for each subject. Source df Sum of Squares Mean Square F Ads 1000 Gender 500 Ads*Gender 1200 Error 54 --- Total 59 5400 --- --- a) Complete the ANOVA table b) Test whether the ad effects differ among the genders (and vice versa) ( =0.05). i. Test Statistic: ii. Conclude interaction exists if test statistic is iii. P-value is above or below 0.05 (circle one)
Q.3. The following partial ANOVA table was obtained from a 2-way ANOVA where 4 newspaper editorials regarding a statewide referendum were to be compared. A total of 60 Republicans, 60 Democrats, and 60 Independents were sampled, and 15 of each were exposed to editorials 1, 2, 3, and 4, respectively. A numeric measure of attitude toward the referendum was obtained for each subject. Source df Sum of Squares Mean Square F Editorials 3600 Political Party 4000 Editorial*Party 7200 Error --- Total 179 23200 --- --- Complete the ANOVA table Test whether the editorial effects differ among the parties (and vice versa) ( =0.05). Test Statistic: Conclude interaction exists if test statistic is P-value is above or below 0.05 (circle one) Q.4. A study is conducted as a 2-Factor ANOVA, with each factor at 3 levels. The Mean Square error was computed to be 180, and each cell of the table is based on 10 replicates (n=10). Obtain the minimum significant difference for comparing means for levels of Factor A (or, equivalently B), based on Bonferroni s method with experiment-wise error rate of 0.05, by completing the following parts (assume no interaction exists): a) Number of comparisons among levels of Factor A b) Critical t-value for simultaneous comparisons c) Standard error of difference between means of 2 levels of Factor A: d) Bonferroni minimum significant difference (part you d add and subtract from estimated difference between level means):
Q.5.A study was conducted to compare the energy efficiencies among a=3 clothing types b=4 dryer types. There were 3 replicates for each combination of clothing type and dryer type. The cell means and marginal means are given below. Clothes\Dryer 1 2 3 4 Overall 1 1.179 1.248 2.180 1.557 1.541 2 1.439 1.437 2.302 1.669 1.712 3 1.292 1.346 2.283 1.536 1.614 Overall 1.303 1.344 2.255 1.587 1.622 p.5.a. Complete the following ANOVA table. Source df SS MS F_obs F(0.05) Treatments 5.442 #N/A #N/A #N/A Clothes 0.176 Dryers 5.230 Clothes*Dryers Error #N/A #N/A Total 5.601 #N/A #N/A #N/A p.5.b. Use Bonferroni s method to compare all pairs of Dryer types. Q.6.: An experiment to measure the effects of 2 factors on dye color permanence in fabrics. The researcher is interested in 3 varieties of cotton (Factor A) and 4 brands of detergent (Factor B). He bases his analysis on a 3 replicates for each combination of cotton and detergent, and the response measured is the level of color in fabric specimen after 10 wash cycles. p.6.a. Complete the ANOVA table, and all tests at the = 0.05 significance level. Source df SS MS F_obs F(0.05) Fabric (A) 800 Detergent (B) 1200 Interaction (AB) 600 Error Total 5000 p.6.b. i. p.6.b. ii. p.6.b. iii. Is there a significant interaction between fabric and detergent? Yes or No Is there a significant main effect for fabric? Yes or No Is there a significant main effect for detergent? Yes or No p.6.c. Compute Bonferroni s Minimum Significant Differences based on an experiment-wise error rate of E = 0.05 (assuming no interaction): p.6.c.i.: For Fabrics: p.6.c.ii: For Detergents:
Q.7. A hotel is interested in studying the effects of washing machines and detergents on whiteness of bed sheets. The hotel has 4 washing machines and 3 brands of detergent. They randomly assign n=4 sheets for each combination of machine and detergent (each sheet is only observed for one combination of machine and detergent). After washing, the sheets are measured for whiteness (high scores are better). Washer\Detergent 1 2 3 Mean 1 25 30 35 30 2 20 35 65 40 3 25 45 50 40 4 30 50 70 50 Mean 25 40 55 40 Source df SS MS F F(0.05) Machine Detergent M*D Error 36000 #N/A #N/A Total 57000 #N/A #N/A #N/A p.8.a. Complete the ANOVA table. p.8.b. Is there a significant interaction between Machine and Detergent on whiteness scores? Yes / No p.8.c. Is there a significant main effect for Machines? Yes / No p.8.d. Is there a significant main effect for Detergents? Yes / No
Q.9. A company productivity office is interested in the effects of lighting and music on employee productivity. Factor A (lighting) had 2 levels (soft and bright) and Factor B (music) had 3 levels (jazz, easy listening, and pop). They selected 30 offices within the firm, and randomly assigned them so that n = 5 received each of the 6 combinations of lighting and music. The response measured was the total output for each office. The summary data are given below. Mean Productivity ANOVA Lighting\Music Jazz Easy Pop Overall Source df SS MS F F(0.05) Soft 30 40 50 40 Lighting Bright 20 40 60 40 Music Overall 25 40 55 40 L*M 5000 Error 48000 #N/A #N/A Total 57500 #N/A #N/A #N/A p.9.a. Complete the ANOVA table. p.9.c. Assuming the interaction is not significant, use Bonferroni s method to compare all 3 pairs of music types. Which types (if any) are significantly different? Q.10. A study is conducted to compare the effects of 4 types of feed for horses. The researcher also believes that the three breeds she works with may differ in terms of weight gains. She samples 12 colts from each breed, randomly assigning 3 to each diet. She measures weight gain on each colt over a 2 month period. Complete the following ANOVA table and conduct the following tests (at 0.05 significance level) Source df SS MS F F(.05) Feeds 3000 Breeds 1000 Interaction 1000 Error Total 35 10000 H 0 : No Feed/Breed Interaction H A : Interaction Present Reject / Don t Reject H 0 H 0 : No Feed Main Effects H A : Feed Main Effects Reject / Don t Reject H 0 H 0 : No Breed Main Effects H A : Breed Main Effects Reject / Don t Reject H 0