Question

1.  An investigator wishes to compare the average time to relief of headache pain under three

distinct medications, call them Drugs A, B, and C. Fifteen patients who suffer from chronic

headaches are randomly selected for the investigation, and five subjects are randomly assigned

to each treatment. The following data reflect times to relief (in minutes) after taking the

assigned drug.

Test if there is a significant difference in the mean times among three treatments. Use α =

0.05. (apply ANOVA)

Drug A	Drug B	Drug C
30	25	15
35	21	20
40	30	25
25	25	22
35	30	24

a) State the null and alternative hypothesis.

Null: all means are equal

Alternative: Atleast one mean is different

b) State your conclusion about the hypothesis based on the test statistic and critical value

Reject the hypothesis because there is a huge difference between the 3 treatments and their means

c) Perform multiple comparison test to show whether there are significant differences between the drugs

Answer 1

Step 1: State null and alternative hypotheses
H_o: μ₁= μ₂= μ₃
H₁: At least one mean is different from the others


Step 2: Find the critical value
k = 3
N = total number of data values from all groups = 15
d.f.N. = k -1 = 2
d.f.D. = N -1 = 12
α = 0.005
Critical Value = 8.5096

Step 3: Calculate F Test Value
ΣX = Sum of all data values = 165+131+106 = 402
N = 15
Grand Mean = X_GM = 402/15 = 26.8
Calculate between-group variance, denoted by S_B2:

SS_B = 5(33 - 26.8)²+ 5(26.2 - 26.8)²+ 5(21.2 - 26.8)²

SS_B = 350.80 / 2 = 175.40

Calculate within-group variance, denoted by S_W2:

SS_W = (5- 1)(32.5)+ (5- 1)(14.7)+ (5- 1)(15.7)

SS_W = 251.60

N - k = 15 - 3 = 12

S_W2 = SS_W/(N - k) = 251.60/12 = 20.96

Calculate F test value:

F = 175.40 / 20.96 = 8.365

Step 4: Make Decision:
Compare F test value = 8.365 with Critical Value = 8.509
The decision is to accept the null hypothesis since 8.365 < 8.5096

Analysis of Variance Summary Table

Source	Sum of Squares	Degrees of Freedom (df)	Mean Square	F test value
Between	SSB = 350.80	k - 1 = 2	MSB = 175.40	F = 8.3656
Within (error)	SSW = 251.60	N - k = 12	MSW = 20.967
Total	602.40	14