# ANOVA   Source of Variation

 ANOVA Source ofVariation SS df MS F p-value Factor A 33,443.41 3 11,147.80 Factor B 24,232.44 2 12,116.22 Interaction 168,937.71 6 28,156.29 Error 135,865.32 36 3,774.04 Total 362,478.88 47
(a) What kind of ANOVA is this?
 One-factor ANOVA Two-factor ANOVA withreplication Two-factor ANOVA withoutreplication
 (b) Calculate each F test statistic and thep-value for each F test using Excel’s function=F.DIST.RT(F,DF1,DF2). (Round yourFcalc values to 3 decimal places andp-values to 4 decimal places.)

 Source ofVariation Fcalc p-value Factor A Factor B Interaction
(c-1)

There is a significant effect due to Factor A.

 True False

(c-2)

There is a significant effect due to Factor B.

 False True

(c-3)

There is a significant interaction between the two factors.

 True False

(a) What Kind of ANOVA is this

Ans : Two-factor ANOVA with replication

In this We are studying two factors and their interaction.

Factor A has four levels

(For Factor A :As degrees of freedom(df)=number of levels -1;Number of levels=df+1=3+1=4)

Factor B has three Levels

(For Factor B: As degrees of freedom(df)=number of levels -1;Number of levels=df+1=2+1=3)

Number of observations for one replication = Number of levelsfor Factor A x Number of levels for Factor B = 4×3 = 12

Degrees of Freedom for Total= Total number of observations -1;

Total number of observations = Degrees of Freedom for Total + 1= 47+1 = 48

Total number of replications = Total number of observations /Number of observations for one replication = 48/12 = 4

(b)

 Source of Variation Fcalc F calc DF1 DF2 p-value p-value Factor A Mean Sum of squares for Factor ‘A’ / Mean sum of squares forError = 11,147.80/ 3,774.04 2.9538 DF for Factor A = 3 DF for Error = 36 F.DIST.RT(2.9538,3,36) 0.04542 Factor B Mean Sum of squares for Factor ‘B’ / Mean sum of squares forError = 12,116.22/ 3,774.04 3.2104 DF for Factor B = 2 DF for Error = 36 F.DIST.RT(3.2104,2,36) 0.05212 Interaction Mean Sum of squares for Interaction / Mean sum of squares forError = 28,156.29/ 3,774.04 7.4605 DF for Interaction = 6 DF for Error = 36 F.DIST.RT(7.4605,6,36) 0.00003
 Source of Variation F calc p-value Factor A 2.9538 0.04542 Factor B 3.2104 0.05212 Interaction 7.4605 0.00003

If p-value is less than level of significance : ; Reject null hypothesis i.e. Factor is significant(Nullhypothesis Factor is not significant). Else Fail to reject nullhypothesis.

(c-1) If level of significance : = 0.05

Factor A : p-value : 0.0454 < 0.05; Reject null hypothesis;Factor ‘A’ is significant; Significant effect due to factor ‘A’ :TRUE

If level of significance : = 0.01

Factor A : p-value : 0.0454 > 0.01; Fail to reject nullhypothesis. Factor ‘A’ is not significant; Significant effect dueto factor ‘A’ : FALSE

(c-2) If level of significance : = 0.05

Factor B : p-value : 0.0521 > 0.05; Fail to reject nullhypothesis; Factor ‘B’ is not significant; Significant effect dueto factor ‘B’ : FALSE

If level of significance : = 0.01

Factor B : p-value : 0.0521 > 0.01; Fail to reject nullhypothesis; Factor ‘B’ is not significant; Significant effect dueto factor ‘B’ : FALSE

(c-3) If level of significance : = 0.05

Interaction : p-value : 0.00003 < 0.05; Reject nullhypothesis; Interaction is significant; Significant effect due toInteraction : True

If level of significance : = 0.01

Interaction : p-value : 0.00003 > 0.01; Reject nullhypothesis; Interaction is significant; Significant effect due toInteraction : True

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