Question

A researcher hypothesizes that students’ performance in a test is better in situations where the surrounding noise is kept lowest. She collects data on her students under three different levels of noise. A higher score indicates better performance. Her data follow. SHOW ALL YOUR WORK.

High Noise	Mod Noise	Low Noise
3	4	6	G =
5	6	8	ΣX2 =	393
4	4	6	N =
3	2	7	k =
2	3	8
T = 17	T = 19	T =
M = 3.4	M = 3.8	M =
SS = 5.2	SS = 8.8	SS =
n = 5	n = 5	n =

1. State the hypotheses

2. Calculate the F ratio and complete the Summary Table.

Source	SS	df	MS	F
Between
Within
Total

3. What is the critical value of F? What decision can you make? What does that indicate?

4. Calculate Turkey’s HSD and determine which pairwise comparisons are statistically significant.

Answer 1

treatment	A1	A2	A3
count, ni =	5	5	5
mean , x̅ i =	3.400	3.80	7.000
std. dev., si =	1.140	1.483	1.000
sample variances, si^2 =	1.300	2.200	1.000
total sum	17	19	35		71	(grand sum)
grand mean , x̅̅ =	Σni*x̅i/Σni =	4.73
square of deviation of sample mean from grand mean,( x̅ - x̅̅)²	1.778	0.871	5.138
					TOTAL
SS(between)= SSB = Σn( x̅ - x̅̅)² =	8.889	4.356	25.689		38.93333333
SS(within ) = SSW = Σ(n-1)s² =	5.200	8.800	4.000		18.0000

no. of treatment , k =   3
df between = k-1 =    2
N = Σn =   15
df within = N-k =   12
  
mean square between groups , MSB = SSB/k-1 =    19.4667
  
mean square within groups , MSW = SSW/N-k =    1.5000
  
F-stat = MSB/MSW =    12.9778

anova table
	SS	df	MS	F	p-value	F-critical
Between:	38.9333	2	19.467	12.978	0.0010	3.885
Within:	18.0000	12	1.500
Total:	56.9333	14
					α =	0.05

critical F = 3.885

reject the null hypothesis

conclusion :    there is enough evidence of significant mean difference among three treatments

==========

tukey's test

Level of significance	0.05
no. of treatments,k	3
DF error =N-k=	12
MSE =	1.5000
q-statistic value(α,k,N-k)	3.7728

critical value = q*√(MSE/2*(1/ni+1/nj)) = 2.07

population mean difference	critical value			result
µ1-µ2	0.40	2.07			means are not different
µ1-µ3	3.600	2.07			means are different
µ2-µ3	3.20	2.07			means are different