Question

Dr. Bose is a psychoacoustician who works in a lab at a headphone company. One of her jobs is to assess new headphone designs and to determine listener preferences. She compared three new headphones that used new noise reduction algorithms (NR1, NR2, & NR3) to a headphone that used the company's current noise reduction algorithm (Current). 32 participants were randomly assigned to one of the four listening conditions. Participants listened to 30 1-minute samples of music from assorted genres and rated the reproduction quality of each sample using a 9-point rating scale (1=poor quality sound; 9=excellent sound). The mean ratings are shown on the right. Did Dr. Bose find any differences in listeners' ratings of music quality among the different noise reduction algorithms?

	Current	NR1	NR2	NR3
1	4.1	2.9	3.3	3.6
2	4.4	4.6	5.2	7.0
3	3.2	1.9	3.7	7.8
4	4.9	3.5	6.1	6.7
5	5.3	3.7	4.7	5.1
6	2.6	7.1	7.2	6.4
7	4.4	3.6	4.1	7.9
8	3.9	3.2	7.3	6.5

Answer 1

We solve the problem using One-way ANOVA technique.

Here the factor of variation is noise reduction algorithms (Current, NR1, NR2, NR3)

Let

- Fixed effect due to the 'Current' noise reduction algorithm

- Fixed effect due to ith noise reduction algorithm (NR1, NR2, NR3) i = 1,2,3

We set up the hypothesis as

Null Hypothesis:

Alternative Hypothesis:

At least two of the means are not equal

Let   , be the jth mean rating of ith noise reduction algorithm

Raw Sum of Squares (R.S.S)

Correction Factor (C.F.)

Total Sum of Squares (T.S.S)

Treatment Sum of Squares (SSt)

From the given data

Error Sum of Squares (SSe)

Constructing ANOVA Table

Source of Variation	SS	df	MS	F
Noise reduction algorithm	32.6809	4-1 = 3
Error	52.5438	31-3 = 28
Total	85.2247	32-1 = 31

F critical value at (3, 28) degrees of freedom and 5% level of significance, F-critical = 2.9467

Since the calculated F (=5.8051) > F-critical (=2.9467) at 5% level of significance we reject the stated null hypothesis and conclude that there are differences in listeners' ratings of music quality among the different noise reduction algorithms