Chap012 Anova (ANALYSIS OF VARIANCE)
Chap012 Anova (ANALYSIS OF VARIANCE)
Chap012 Anova (ANALYSIS OF VARIANCE)
Chapter 12
McGraw-Hill/Irwin Copyright © 2010 by The McGraw-Hill Companies, Inc. All rights reserved.
GOALS
12-2
The F Distribution
It is
– used to test whether two samples are from populations having equal
variances
– applied when we want to compare several population means
simultaneously. The simultaneous comparison of several population
means is called analysis of variance(ANOVA).
– In both of these situations, the populations must follow a normal
distribution, and the data must be at least interval-scale.
12-3
Characteristics of F-Distribution
1. There is a “family” of F
Distributions. A particular
member of the family is
determined by two
parameters: the degrees of
freedom in the numerator and
the degrees of freedom in the
denominator.
2. The F distribution is
continuous
3. F cannot be negative.
4. The F distribution is positively
skewed.
5. It is asymptotic. As F the
curve approaches the X-axis
but never touches it.
12-4
Comparing Two Population Variances
The F distribution is used to test the hypothesis that the variance of one
normal population equals the variance of another normal population.
Examples:
Two Barth shearing machines are set to produce steel bars of the same length.
The bars, therefore, should have the same mean length. We want to ensure
that in addition to having the same mean length they also have similar variation.
The mean rate of return on two types of common stock may be the same, but
there may be more variation in the rate of return in one than the other. A
sample of 10 technology and 10 utility stocks shows the same mean rate of
return, but there is likely more variation in the Internet stocks.
A study by the marketing department for a large newspaper found that men and
women spent about the same amount of time per day reading the paper.
However, the same report indicated there was nearly twice as much variation in
time spent per day among the men than the women.
12-5
Test for Equal Variances
12-6
Test for Equal Variances - Example
12-7
Test for Equal Variances - Example
12-8
Test for Equal Variances - Example
12-9
Test for Equal Variances - Example
We conclude that there is a difference in the variation of the travel times along
12-10
the two routes.
Test for Equal Variances – Excel
Example
12-11
Comparing Means of Two or More
Populations
Assumptions:
– The sampled populations follow the normal
distribution.
– The populations have equal standard deviations.
– The samples are randomly selected and are
independent.
12-12
Comparing Means of Two or More
Populations
H0: µ1 = µ2 =…= µk
H1: The means are not all equal
Reject H0 if F > F,k-1,n-k
12-13
Analysis of Variance – F statistic
SST k 1
F
SSE n k
12-14
Comparing Means of Two or More
Populations – Illustrative Example
Joyce Kuhlman manages a regional
financial center. She wishes to
compare the productivity, as
measured by the number of
customers served, among three
employees. Four days are randomly
selected and the number of
customers served by each employee
is recorded. The results are:
12-15
Comparing Means of Two or More
Populations – Example
12-16
Comparing Means of Two or More
Populations – Example
H0: µE = µA = µT = µO
H1: The means are not all equal
Reject H0 if F > F,k-1,n-k
Step 2: State the level of significance.
The .01 significance level is stated in the problem.
12-17
Comparing Means of Two or More
Populations – Example
12-18
Comparing Means of Two or More
Populations – Example
12-19
Comparing Means of Two or More
Populations – Example
12-20
Computing SS Total and SSE
12-21
Computing SST
The computed value of F is 8.99, which is greater than the critical value of 5.09,
so the null hypothesis is rejected.
Conclusion: The population means are not all equal. The mean scores are not
the same for the four airlines; at this point we can only conclude there is a
difference in the treatment means. We cannot determine which treatment groups
differ or how many treatment groups differ.
12-22
Confidence Interval for the
Difference Between Two Means
When we reject the null hypothesis that the means are equal, we may
want to know which treatment means differ. One of the simplest
procedures is through the use of confidence intervals.
1 1
X X2 t MSE
1
2
1
n n
12-23
Confidence Interval for the
Difference Between Two Means - Example
12-25
Two-Way Analysis of Variance
SSB k( x b x G ) 2
12-26
Two-Way Analysis of Variance - Example
WARTA conducted several tests to determine whether there was a difference in the
mean travel times along the four routes. Because there will be many different drivers,
the test was set up so each driver drove along each of the four routes. Next slide shows
the travel time, in minutes, for each driver-route combination. At the .05 significance
level, is there a difference in the mean travel time along the four routes? If we remove
the effect of the drivers, is there a difference in the mean travel time?
12-27
Two-Way Analysis of Variance - Example
Sample Data
12-28
Two-Way Analysis of Variance - Example
SSB k( x b x G ) 2
12-30
Two-Way Analysis of Variance - Example
12-31
Two-Way Analysis of Variance – Excel
Example
12-34
Graphical Observation of Mean Times
12-36
12-37
Three Tests in ANOVA with Replication
12-38
ANOVA Table
12-39
Excel Output
Driver Route
12-40
One-way ANOVA for Each Driver
12-41