Question

please show work for each!

ΑΙ Factor B B. B2 T = 40 T = 10 M=4 M= 1 SS-50 SS = 30 Factor A A2 T-50 M=5 SS - 60 T = 20 M = 2 SS = 40 N-40 G = 120 EX =640 Use a two-way ANOVA with a=0.05 to evaluate the main effects and the interaction. m Source SS df MS F Between treatments | 1 6.67 I 2.00 Factor A (ROWS) 18.00 Factor B (COLUMNS) Interaction (AXB) 0.00 Within treatment Total a. What critical value should be used to evaluate the main effect of Factor A? b. Was there a main effect of Factor A? c. What critical value should be used to evaluate the main effect of Factor B? d. Was there a main effect of Factor B? e. What critical value should be used to evaluate the interaction? f. Was there an interaction?

Answer 1

Σx2. SSTotal = 640 – 120/40 = 280 N with degrees of freedom (df) = 40 - 1 = 39 SSError 50 +30 + 60 + 40 = 180 with df = 39 – 3 = 36 SSBetween SSTotal – SSError = 280 – 180 100 with df = 3 SSA = 50/20 + 70/20 - 120P/40 = 10 with df = 1 SSB = 902/20+ 302/20 - 120/40 = 90 with df = 1 SS(A x B) SS Between - SSA - SSB 0 with df = 1

ANOVA table:

Source	SS	df	MS=SS/df	F
Between treatments	100	3	100/3	6.67
A	10	1	10	2
B	90	1	90	18
AxB	0	1	0	0
Within treatment	180	36	180/36=5
Total	280	39

$Degrees~of~freedom~(df)~for~Main~effect~of~Factor~A=2-1=1\\ Degrees~of~freedom~(df)~for~Main~effect~of~Factor~B=2-1=1\\ Degrees~of~freedom~(df)~for~interaction~effect~A\times B=(2-1)(2-1)=1\\ Degrees~of~freedom~(df)~for~error=(40-1)-3=36.\\\\ a.~Critical~value~of~A=F_{0.05,1,36}=4.1132\\\\ (R~code:~round(qf(1-0.05,1,36),4)).\\\\ b.~Since~F-ratio~corresponding~A=2.00<4.1132,~main~effect~of~factor~A~is\\ insignificant.\\\\ c.~Critical~value~of~B=F_{0.05,1,36}=4.1132\\\\ d.~Since~F-ratio~corresponding~B=18.00>4.1132,~main~effect~of~factor~B~is\\ significant.\\\\ e.~Critical~value~of~A\times B=F_{0.05,1,36}=4.1132\\\\ f.~Since~F-ratio~corresponding~A\times B=0.00<4.1132,~interaction~effect~is\\ insignificant.\\\\$