Question

MA1 = 102.5

MA2 = 103

MA3 = 104.5

Source			SS	df	MS	F
Factor A			17.333	2	8.667	7.00
Persons			16.667	7	2.380953
Error (within)			17.333	14	1.238
Total			51.33	23

• With α = .05, is there a statistically significant difference between the means of caffeine consumption?  
• If so, what is the value of η²?
• If applicable, using the post hoc test of your choice (and showing your work), which means are different?

Answer 1

Null Hypothesis H0: The means of caffeine consumptions are equal for all groups.

Alternative Hypothesis Ha: Not all of the means of caffeine consumptions are equal.

F Test statistic for Factor A = 7.00

Critical value of F at α = .05 and df = 2, 14 is 3.74

Since the observed test statistic (7.00) is greater than the critical value (3.64) , we reject H0 and conclude that there is statistically significant difference between the means of caffeine consumption.

η2 = SS Factor A / SS Total = 17.333 / 51.33 =  0.3377

Using Tukey's PostHoc test,

Tukey HSD is calculated using the below formula

where   is a critical value of the studentized range for α, the number of treatments or samples r, and the within-groups degrees of freedom . We get this value from studentized range table.

is the within groups mean square from the ANOVA table and n is the sample size for each treatment.

N = df Total + 1 = 23 + 1 = 24

n = N / r = 24 / 3 = 8

From Anova table,

= 2 , = 14

n = 8, r = 3, α = 0.05

From studentized range table, = 3.701

So,

So, any aboslute mean difference greater than 1.8505 is significantly different.

A2 - A1 = 103 - 102.5 = 0.5

A3 - A1 = 104.5 - 102.5 = 2

A3 - A2 = 104.5 - 103 = 1.5

Only the difference between A1 and A3 is greater than the critical Tukey HSD value. Thus, the means of A1 and A3 are significanlty different at α = .05