ANALYSIS OF VARIANCE EXAMPLE

ANALYSIS OF VARIANCE

Many businesses have music piped into the work areas to improve the environment. At a company an experiment is performed to compare different types of music. Three types of music – country, rock, and classical – are tried, each on four randomly selected days. Each day the productivity, measured by the number of items produced, is recorded. The results appear below.

A. Can we conclude from this information that the mean number of items produced differs for at least two of the three types of music? Use a = .05.

B. Carefully explain what the p-value found in part A means.

C. Which type(s) of music seem to be best?

D. Which type(s) of music seem to be worst?

SOLUTION

A. The parameters of interest are m₁ , the mean number of items produced for all days when country music is played; m₂, the mean number of items produced for all days when rock music is played; and m₃, the mean number of items produced for all days when classical music is played.

H₀: m₁ = m₂ = m₃ H_a: At least two of the means are not equal

Decision Rule: Accept H_a if the calculated p-value < .05.

Calculations from StatCrunch: F = 10.66, p-value = 0.0042 < .05 ---> Accept H_a

Interpretation: At the .05 level of significance I conclude that the mean number

B. If the mean productivity were the same for all three types of music (the null hypothesis is true), then the probability of observing three sample means as varied, or more varied, as those obtained in this experiment is 0.0042. Therefore it is extremely unlikely (smaller than α = .05) that the sample means would have such diverse values if all the population means are equal. This is why the alternative hypothesis was accepted.

C. The results of the multiple comparison tests of the mean numbers of items produced for each of the three types of music are shown below. Interpretations are listed in terms of which is largest because "best", in this application, means largest.

Comparison	Value of t	p-value	Interpretation
Country vs. Rock	2.81	0.020	Country > Rock
Country vs. Classical	-1.77	0.110	NS
Rock vs. Classical	-4.58	0.001	Classical > Rock

From these analyses we see that rock music is certainly not the best in terms of worker productivity. Classical music may be best, but perhaps country music could be.

D. From these same analyses we see that rock music is the worst in terms of worker productivity.

COMMENTS ABOUT THE SOLUTION

The usual structure for presenting the solution to a test of hypothesis was followed. All test of hypotheses solutions should follow this structure.

This procedure is valid when all three samples are random and independently selected. All three populations must be normally distributed with equal variances.

The complete StatCrunch analyses are below.

Analysis of Variance results:
Responses stored in Items.
Factors stored in Music.

Factor means ANOVA table

Music

n

Mean

Std. Error

Classical

4

854.25

12.243196

Country

4

823.75

15.228127

Rock

4

775.25

8.045444

Source

df

SS

MS

F-Stat

P-value

Treatments

2

12698

6349

10.664116

0.0042

Error

9

5358.25

595.3611

Total

11

18056.25

A graph of the appropriate F distribution and the resulting p-value is furnished below.

Music	n	Mean	Std. Error
Classical	4	854.25	12.243196
Country	4	823.75	15.228127
Rock	4	775.25	8.045444

Source	df	SS	MS	F-Stat	P-value
Treatments	2	12698	6349	10.664116	0.0042
Error	9	5358.25	595.3611
Total	11	18056.25