Download STAT 500 Solutions Practice Problems: ANOVA

STAT 500
Solutions Practice Problems: ANOVA
1. To compare the heights of Soprano, Alto, Tenor and Bass singers, data have been collected
from the NY Choral Society in 1979 (singers.txt). Determine whether these four groups have the
same mean height. Use α=0.05. (Source: DASL)
We use ANOVA to answer this question as we have a response variable (heights) measured
across a categorical variable (Singers) of more than two levels (Soprano, Alto, Tenor, and
Bass).
Ho: is that the mean heights of the four types of singers are equal. In statistical notation
this is: µS = µA = µT = µB
Ha: is that at least one of these means differ, or can be written as not all the means are
equal.
With a large sample size we are not overly concerned with normality, but since the sample
sizes differ across the groups (36, 35, 20, and 39, respectively) we would be concerned if the
variances were not equal. From Stat > ANOVA > Test for equal variances using
“Response data are in separate column for each factor level” we get the following. This
indicates a violation of the equal variance assumption and could allow one to question the
reliability of our results.
Test for Equal Variances: Soprano, Alto, Tenor, Bass
Multiple comparison intervals for the standard deviation, α = 0.05
Multiple Comparisons
P-Value
Soprano
0.031
Levene’s Test
P-Value
0.020
Alto
Tenor
Bass
1.5
2.0
2.5
3.0
3.5
4.0
4.5
5.0
If intervals do not overlap, the corresponding stdevs are significantly different.
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To continue with the ANOVA analysis in Minitab selecting Stat > ANOVA > One-Way use
choose that the “Response data are in separate column for each factor level.” We also
check the Comparisons tab and check the box for Tukey. This results in the following
output and multiple comparison graph.
One-way ANOVA: Soprano, Alto, Tenor, Bass
Method
Null hypothesis
Alternative hypothesis
Significance level
Rows unused
All means are equal
At least one mean is different
α = 0.05
26
Equal variances were assumed for the analysis.
Factor Information
Factor
Factor
Levels
4
Values
Soprano, Alto, Tenor, Bass
Analysis of Variance
Source
Factor
Error
Total
DF
3
126
129
Adj SS
1058.5
796.7
1855.3
Adj MS
352.843
6.323
F-Value
55.80
P-Value
0.000
Model Summary
S
2.51462
R-sq
57.06%
R-sq(adj)
56.03%
R-sq(pred)
54.06%
Means
Factor
Soprano
Alto
Tenor
Bass
N
36
35
20
39
Mean
64.250
64.886
69.150
70.718
StDev
1.873
2.795
3.216
2.361
95%
(63.421,
(64.045,
(68.037,
(69.921,
CI
65.079)
65.727)
70.263)
71.515)
Pooled StDev = 2.51462
Tukey Pairwise Comparisons
Grouping Information Using the Tukey Method and 95% Confidence
Factor
Bass
Tenor
Alto
Soprano
N
39
20
35
36
Mean
70.718
69.150
64.886
64.250
Grouping
A
A
B
B
Means that do not share a letter are significantly different.
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Tukey Simultaneous 95% CIs
Difference of Means for Soprano, Alto, ...
Alto - Soprano
Tenor - Soprano
Bass - Soprano
Tenor - Alto
Bass - Alto
Bass - Tenor
0
2
4
6
8
If an interval does not contain zero, the corresponding means are significantly different.
From the ANOVA output F test statistic (55.8) and corresponding p-value (less than 0.001),
we would reject the null hypothesis at a 0.05 level of significance. We would conclude that
the sample data provides statistical evidence that there is a difference in the mean heights
across the four types of singers. Since we have a significant result, we can review the
multiple comparison results to determine where differences occur.
From the letter grouping information, we see that Bass and Tenor are not different (both
letter A) and Alto and Soprano are not different (both letter B). However, Bass and Tenor
are different from Alto and Soprano since they do not have a common letter.
From the multiple comparison graph of confidence intervals, we confirm the similarity
between Alto and Soprano (interval includes zero) and Bass and Tenor (interval includes
zero). Furthermore, we can reach the following conclusions in mean heights:
Tenor exceeds Soprano; Bass exceeds Soprano; Tenor exceeds Alto; and Bass exceeds Alto.
All of which reflects the results of the letter grouping.
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