Question

Starbucks Bigby Tim Hortons 1 7 4 1 7. 4 2 8 5 3 8 5 3 8 9 6 3 5 9 6 10 7 6

What is the critical q-value?

What is the critical difference needed?

Which means are significantly different from one another?

Answer 1

treatment	G1	G2	G3	G4	G5
count, ni =	8	8	8
mean , x̅ i =	3.000	8.25	5.38
std. dev., si =	1.8	1.0	1.1
sample variances, si^2 =	3.143	1.071	1.125
total sum	24	66	43				133	(grand sum)
grand mean , x̅̅ =	Σni*x̅i/Σni =	5.54
( x̅ - x̅̅ )²	6.460	7.335	0.028
							TOTAL
SS(between)= SSB = Σn( x̅ - x̅̅)² =	51.681	58.681	0.222				110.5833333
SS(within ) = SSW = Σ(n-1)s² =	22.000	7.500	7.875				37.3750

no. of treatment , k = 3

df between = k-1 = 2

N = Σn = 24

df within = N-k = 21

mean square between groups , MSB = SSB/k-1 = 110.5833/2= 55.2917

mean square within groups , MSW = SSW/N-k = 37.375/21= 1.7798

F-stat = MSB/MSW = 55.2917/1.7798= 31.0669

anova table
	SS	df	MS	F	p-value	F-critical
Between:	110.58	2.00	55.29	31.07	0.000	3.467
Within:	37.38	21.00	1.78
Total:	147.96	23.00
					α =	0.05

F stat = 31.07

critical value at 0.05 = 3.467

f stat > critical value , reject Ho

................

TUKEY HSD/KRAMER
Level of significance	0.05
no. of treatments,k	3
DF error =N-k=	21
MSE =	1.7798
q-critical value(0.05,3,21)=	3.5646

critical value = q*√(MSE/2*(1/ni+1/nj))  
=1.68

confidence interval = mean difference ± critical value      

           
if confidence interval do not contains zero, then means are significantly different.                  
critical value = q*√(MSE/n)                  

if absolute difference of means > critical value,means are significnantly different ,otherwise not                  

					confidence interval
	mean difference	absolute mean difference		critical value	lower limit	upper limit			result
µ1-µ2	-5.25	5.25		1.68	-6.93	-3.57			yes, means are different
µ1-µ3	-2.38	2.38		1.68	-4.06	-0.69			yes, means are different
µ2-µ3	2.88	2.88		1.68	1.19	4.56			yes, means are different

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