Question

Problem: Susan Sound predicts that students will learn most effectively with a constant background sound, as opposed to an unpredictable sound or no sound at all. She randomly divides twenty-four students into three groups of eight. All students study a passage of text for 30 minutes. Those in group 1 study with background sound at a constant volume in the background. Those in group 2 study with noise that changes volume periodically. Those in group 3 study with no sound at all. After studying, all students take a 10 point multiple choice test over the material. Their scores follow:
group test scores

1) constant sound 7 4 6 8 6 6 2 9
2) random sound 5 5 3 4 4 7 2 2
3) no sound 2 4 7 1 2 1 5 5

Please give me step by step how to answer the question.

Answer 1

The given problem is One Way ANOVA problem.

The null and alternate hypothesis
H₀: μ₁ = μ₂ = μ₃
H_a: μ₁ ≠ μ₂ ≠ μ₃

Calculate the Sum of Squares

	Treatments
	Constant Sound	Random Sound	No Sound
	7	5	2
	4	5	4
	6	3	7
	8	4	1
	6	4	2
	6	7	1
	2	2	5
	9	2	5
				Total
Ti	48	32	27	GT=107
∑X2	322	148	125	R.S.S = 595

Correction factor (C.F.) =

Total S.S. = R.S.S - C.F. = 595 - 477.0417 =117.9583

S.S.T (Between treatments ) =

S.S.E. ( within treatments) = T.S.S. - S.S.T. = 117.9583 - 30.0833 = 87.875

Calculate the Degrees of Freedom

df_total =        n – 1 = 24 – 1 = 23
df_within = n – k = 24 – 3 = 21
df_between = k – 1 = 3 – 1 = 2

Calculate the Mean Squares

ANOVA Table
Source	SS	df	MS	Variance ratio
Between-treatments	30.0833	2	15.0417	F = 3.5946
Within-treatments	87.875	21	4.1845
Total	117.9583	23

From the table of F distribution, the critical value of F for 0.05 significance and degrees of freedom of(df1 = 2 and df2 = 21) we have:

F = 3.4668

Since the calculated(absolute value) of F is greater than the tabulated value, we reject the null hypothesis and conclude that at least two of the means are significantly different from each other.