A randomized block design has three levels of factor A and five levels of factor
ID: 3379720 • Letter: A
Question
A randomized block design has three levels of factor A and five levels of factor B where six replicates for each combination are examined. The results include the following sum of square terms: Construct an ANOVA table. (Leave no cells blank - be certain to enter "0" wherever required. Round "p" value to 3 decimal places and all other answers except "df" and "SS" to 2 decimal places.) At the 1% significance level, can you conclude that there is interaction between factor A and factor B? At the 1% significance level, can you conclude that the factor A means differ? At the 1% significance level, can you conclude that the factor B means differ? Entries in this table provide the values of that correspond to a given upper-tail area alpha and a specified number of degrees of freedom in the numerator df_1 and degrees of freedom in the denominator df_2. For example, for alpha = 0.05, df_1 = 8, and df_2 = 6Explanation / Answer
ANOVA Sources of variation
Sum of squares
degrees of freedom
Mean Sum of Squares
F
F critical at 1%
Rows
SSA=1042
p-1=2
MSA=SSA/(p-1)
=521
FA=MSA/MSE
=521/1.3733
=379.378
F0.01,2,75 =4.89988
Columns
SSB=358
q-1=4
MSB=SSB/(q-1)
=89.5
FB=MSB/MSE
=89.5/1.3733
=65.171
F0.01,4,75 =3.58011
Interaction
SSAB=39
(p-1)*(q-1)=8
MSAB=SSAB/
(p-1)*(q-1)
=4.875
FAB =MSAB/MSE
=4.875/1.3733
=3.5498
F0.01,8,75 =2.75804
Error
SSE=103
p*q*(r-1)=75
MSE=SSE/(p*q*(r-1)) =1.3733
Total
SST=1542
p*q*r-1=89
Here p=no of levels of factor A
q=no of levels of factor B
r=no. of replicates
p=3,q=5,r=6
b)
Here we are interested in testing H0:There is any interaction between the factors A and B against H1:not H0
In the light of the given observations at 1% level of significance we reject H0 and conclude that there is no interaction between A and B since observed(FAB)>F0.01,8,75 .
c)At 1% level of significance we cannot conclude about the means of the factor A
d)At 1% level of significance we cannot conclude about the means of the factor B
since the test for interaction effect is rejected.
ANOVA Sources of variation
Sum of squares
degrees of freedom
Mean Sum of Squares
F
F critical at 1%
Rows
SSA=1042
p-1=2
MSA=SSA/(p-1)
=521
FA=MSA/MSE
=521/1.3733
=379.378
F0.01,2,75 =4.89988
Columns
SSB=358
q-1=4
MSB=SSB/(q-1)
=89.5
FB=MSB/MSE
=89.5/1.3733
=65.171
F0.01,4,75 =3.58011
Interaction
SSAB=39
(p-1)*(q-1)=8
MSAB=SSAB/
(p-1)*(q-1)
=4.875
FAB =MSAB/MSE
=4.875/1.3733
=3.5498
F0.01,8,75 =2.75804
Error
SSE=103
p*q*(r-1)=75
MSE=SSE/(p*q*(r-1)) =1.3733
Total
SST=1542
p*q*r-1=89
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