1 answer

What is the critical q-value? What is the critical difference needed? Which means are significantly different...

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?

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

Answers

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

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

Please let me know in case of any doubt.

Thanks in advance!


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