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Chapter 16
Analysis
of
Variance
Copyright ©2011 Brooks/Cole, Cengage Learning
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9.1 Parameters, Statistics,
and Statistical Inference
A statistic is a numerical value computed from a
sample. Its value may differ for different samples.
e.g. sample mean x , sample standard deviation s,
and sample proportion p̂.
A parameter is a numerical value associated with
a population. Considered fixed and unchanging.
e.g. population mean m, population standard
deviation s, and population proportion p.
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ANOVA
Analysis of variance: tool for analyzing
how the mean value of a quantitative
response variable is affected by one or more
categorical explanatory factors.
If one categorical variable: one-way ANOVA
If two categorical variables: two-way ANOVA
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16.1 Comparing Means
with an ANOVA F-Test
H0: m1 = m2 = … = mk
Ha: The population means are not all equal.
F-statistic:
Variation among sample means
F
Natural variation within groups
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Variation among sample means
F
Natural variation within groups
Variation among sample means is 0 if all
k sample means are equal and gets larger
the more spread out they are.
If large enough  evidence at least one
population mean is different from others
 reject null hypothesis.
p-value found using an F-distribution (more later)
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Example 16.1 Seat Location and GPA
Q: Do best students sit in the front of a classroom?
Data on seat location and GPA for n = 384 students;
88 sit in front, 218 in middle, 78 in back
Students sitting in the front
generally have slightly
higher GPAs than others.
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Example 16.1 Seat Location and GPA
H0: m1 = m2 = m3
Ha: The three population means are not all equal.
The F-statistic is 6.69 and the p-value is 0.0001.
p-value so small  reject H0 and conclude there
are differences among the population means.
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Example 16.1 Seat Location and GPA
95% Confidence Intervals for 3 population means:
Interval for “front” does not overlap with the
other two intervals  significant difference
between mean GPA for front-row sitters and
mean GPA for other students
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Notation for Summary Statistics
k = number of groups
x , si, and ni are the mean, standard deviation,
and sample size for the ith sample group
N = total sample size = n1 + n2 + … + nk
Example 16.2 Seat Location and GPA
Three seat locations  k = 3
n1 = 88, n2 = 218, n3 = 78; N = 88+218+78 = 384
x1  3.2029, x2  2.9853, x3  2.9194
s1  0.5491, s2  0.5577, s3  0.5105
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Assumptions for the F-Test
• Samples are independent random samples.
• Distribution of response variable is a normal curve
within each population.
• Different populations may have different means.
• All populations have same standard deviation, s.
How k = 3 populations might look …
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Conditions for Using the F-Test
• F-statistic can be used if data are not extremely
skewed, there are no extreme outliers, and group
standard deviations are not markedly different.
• Tests based on F-statistic are valid for data with
skewness or outliers if sample sizes are large.
• A rough criterion for standard deviations is that
the largest of the sample standard deviations
should not be more than twice as large as the
smallest of the sample standard deviations.
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Example 16.3 Seat Location and GPA
• The boxplot showed two outliers in the group
of students who typically sit in the middle of
a classroom, but there are 218 students in that
group so these outliers don’t have much
influence on the results.
• The standard deviations for the three groups
are nearly the same.
• Data do not appear to be skewed.
Necessary conditions for F-test seem satisfied.
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The Family of F-Distributions
• Skewed distributions with minimum value of 0.
• Specific F-distribution indicated by two parameters
called degrees of freedom: numerator degrees of
freedom and denominator degrees of freedom.
• In one-way ANOVA,
numerator df = k – 1,
and
denominator df = N – k
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Determining the p-Value
Statistical Software reports the p-value in output.
Table A.4 provides critical values for 1% and 5%
significance levels.
• If the F-statistic is > than the 5% critical value,
the p-value < 0.05.
• If the F-statistic is > than the 1% critical value,
the p-value < 0.01 .
• If the F-statistic is between the 1% and 5% critical
values, the p-value is between 0.01 and 0.05.
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16.2 Details of One-Way
Analysis of Variance
Fundamental concept: the variation among the data
values in the overall sample can be separated into:
(1) differences between group means
(2) natural variation among observations within a group
Total variation =
Variation between groups + Variation within groups
ANOVA Table displays this information.
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Measuring Variation
Between Groups
Sum of squares for groups = SS Groups
SS Groups   groups ni xi  x 
2
Numerator of F-statistic = mean square for groups
SS Groups
MS Groups 
k 1
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Measuring Variation
within Groups
Sum of squared errors = SS Error
SS Errors  groups ni  1si2
Denominator of F-statistic = mean square error
SS Error
MSE 
N k
Pooled standard deviation:
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sp 
MSE
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Measuring Total Variation
Total sum of squares = SS Total = SSTO
SS Total  values xij  x 
2
SS Total = SS Groups + SS Error
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General Format of a
One-Way ANOVA Table
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Example 16.7 Analysis of Variation
among Weight Losses
x1  7
x2  9
x3  15
Program 3 appears to have the highest weight loss overall.
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Example 16.8 Analysis of Variation
among Weight Losses
x1  7, x2  9, x3  15 and x  10
n1  4, n2  3, n3  3 and N  10
SS Groups  groups ni xi  x 
2
 47  10  39  10  315  10  114
2
2
2
SS Groups 114
MS Groups 

 57
k 1
3 1
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Example 16.8 Analysis of Variation
among Weight Losses
x1  7, x2  9, x3  15 and x  10
n1  4, n2  3, n3  3 and N  10
SS Total  values xij  x 
2
 7  10  9  10  5  10  7  10
2
2
2
2
 9  10  11  10  7  10
2
2
2
 15  10  12  10  18  10
 148
2
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2
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Example 16.8 Analysis of Variation
among Weight Losses
x1  7, x2  9, x3  15 and x  10
n1  4, n2  3, n3  3 and N  10
SS Error  SS Total - SS Groups
 148-114  34
SS Error
34
MSE 

 4.857
N k
10  3
MS Groups
57
F

 11.74 with 2 and 7 df
MSE
4.857
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Example 16.8 Analysis of Variation
among Weight Losses
“Factor” used instead of Groups as the groups (weight-loss
programs) form an explanatory factor for the response.
Note: Pooled StDev is s p  MSE  4.86  2.204
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Example 16.9 Top Speeds of Supercars
Data: top speeds for six runs on
each of five supercars. Kitchens (1998, p. 783)
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Example 16.9
Top Speeds
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Example 16.9
Top Speeds
• F = 25.15 and p-value is 0.000  reject null hypothesis
that population mean speeds are same for all five cars.
• Conditions are satisfied. Data not skewed and no
extreme outliers. Largest sample std dev (5.02 Viper) not
more than twice as large as smallest std dev (2.92 Acura).
• MS Error =14.5 is an estimate of variance of top speed for
hypothetical distribution of all possible runs with one car.
Estimated standard deviation for each car is 3.81.
• Based on sample means and CIs: Porsche and Ferrari
seem to be significantly faster than other cars.
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Computation of 95% Confidence
Intervals for the Population Means
In one-way analysis of variance, a confidence
interval for a population mean mi is


s
p
*

xi  t
 n 
 i
where s p 
MSE and
t* is such that the confidence level is the probability
between -t* and t* in a t-distribution with df = N – k.
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16.3 Other Methods
When data are skewed or extreme outliers present
…better to analyze the median instead of mean
H0: Population medians are equal.
Ha: Population medians are not all equal.
Two such tests are:
1. Kruskal-Wallis Test
2. Mood’s Median Test
Also called nonparametric tests.
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Example 16.12 Drinks and Seat Location
Data: Seat location and
number of alcoholic drinks per week
Data appear skewed, sample
standard deviations differ.
Students sitting in the back report drinking more.
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Example 16.12 Drinks and Seat Location
P = 0.000  strong evidence that the population
median number of drinks per week are not all equal.
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Example 16.13 Drinks and Seat Location
P = 0.000 => the null hypothesis of equal
population medians can be rejected.
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