Question

1. (20) Consider the following incomplete ANOVA table from a 2-factor experiment that has been conducted. SS D F p-value Source A MS 50.00 40.00 15.00 B 80.00 30.00 2 Interaction Error | Total 172.00 12 171 a. What are the missing values from the table? (Hint: For the p-values, you can use the right-tail F-distribution function in Excel F.DIST.RT(x, df-numerator, df- denominator, true). b. How many levels were used for factor B? c. How many replications were performed? d. What is your interpretation of the results? e. What is the estimate of the random error component standard deviation?

Answer 1

a) We know      
MS = SS/DF, hence SS = MS*DF     
Thus, SS for A is      
SS = 1*50 = 50     
      
DF = SS/MS     
Thus, DF for B     
DF = 80/40 = 2     
      
Total SS is sum of all SS     
Thus, 172 = 50 + 80 + 30 + Error SS     
SS for Error = 172 - 160     
SS for Error = 12     
      
MS for Error = SS / DF     
   = 12/12 = 1     
      
F for A is MS for A/ MS for Error     
F for A = 50/1 = 50     
      
F for B is MS for B/ MS for Error     
F for B = 40/1 = 40     
      
F for interaction is MS for interaction/ MS for Error     
F for interaction = 15/1 = 15     
      
p-value for A = F.DIST.RT(50, 1, 12)     
   = 0.000013    
      
p-value for B = F.DIST.RT(40, 1, 12)     
   = 0.000038    
      
p-value for interaction = F.DIST.RT(15, 1, 12)     
   = 0.0022    
      
Filled ANOVA table is

Source	SS	DF	MS	F	p-value
A	50	1	50	50	0.000013
B	80	2	40	40	0.000038
Interaction	30	2	15	15	0.0022
Error	12	12	1
Total	172	17

b) Degrees of Freedom for B = 2     
Levels used for factor B = 2 + 1 = 3     
      
c) Levels used for factor A = DF for A + 1 = 1 + 1 = 2     
Total DF + 1 = Levels of A * Levels of B * n          where n is the number of replications     
18 = 2 * 3 * n     
n = 3     
Number of replications performed = 3     
      
d) Assume level of significance = 0.05     
p-value for A is less than 0.05     
Hence there is no effect of factor A     
p-value for B is less than 0.05     
Hence there is no effect of factor B     
p-value for interaction is less than 0.05     
Hence there is no effect of interaction     
      
e) Estimate of the random error component standard deviation     
= squareroot(12)  (Since standard deviation is asked we take the square root of SS for Error)   
= 3.4641