1. Comparing Three or More Means ANOVA (One-Way Analysis of Variance)
Lesson
Comparing Three or More Means ANOVA (One-Way Analysis of Variance)

2. Objectives Verify the requirements to perform a one-way ANOVA
Test a claim regarding three or more means using one way ANOVA

3. Vocabulary
ANOVA – Analysis of Variance: inferential method that is used to test the equality of three or more population means
Robust – small departures from the requirement of normality will not significantly affect the results
Mean squares – is an average of the squared values (for example variance is a mean square)
MST – mean square due to the treatment
MSE – mean square due to error
F-statistic – ration of two mean squares

4. One-way ANOVA Test Requirements
There are k simple random samples from k populations
The k samples are independent of each other; that is, the subjects in one group cannot be related in any way to subjects in a second group
The populations are normally distributed
The populations have the same variance; that is, each treatment group has a population variance σ2

5. ANOVA Requirements Verification
ANOVA is robust, the accuracy of ANOVA is not affected if the populations are somewhat non- normal or do not quite have the same variances
Particularly if the sample sizes are roughly equal
Use normality plots
Verifying equal population variances requirement:
Largest sample standard deviation is no more than two times larger than the smallest

6. ANOVA – Analysis of Variance
Computing the F-test Statistic
1. Compute the sample mean of the combined data set, x
Find the sample mean of each treatment (sample), xi
Find the sample variance of each treatment (sample), si2
Compute the mean square due to treatment, MST
Compute the mean square due to error, MSE
Compute the F-test statistic:
mean square due to treatment MST F = = mean square due to error MSE
ni(xi – x) (ni – 1)si2
MST = MSE =
k – l n – k Σ k Σ k
n = 1
n = 1

7. MSE and MST
MSE - mean square due to error, measures how different the observations, within each sample, are from each other
It compares only observations within the same sample
Larger values correspond to more spread sample means
This mean square is approximately the same as the population variance
MST - mean square due to treatment, measures how different the samples are from each other
It compares the different sample means
Under the null hypothesis, this mean square is approximately the same as the population variance

8. ANOVA – Analysis of Variance Table
Source of Variation
Sum of Squares
Degrees of Freedom
Mean Squares
F-test Statistic
F Critical Value
Treatment
Σ ni(xi – x)2
k - 1
MST
MST/MSE
F α, k-1, n-k
Error
Σ (ni – 1)si2
n - k
MSE
Total
SST + SSE
n - 1

9. Excel ANOVA Output Classical Approach: P-value Approach:
Test statistic > Critical value … reject the null hypothesis
P-value Approach:
P-value < α (0.05) … reject the null hypothesis

10. TI Instructions
Enter each population’s or treatments raw data into a list
Press STAT, highlight TESTS and select F: ANOVA(
Enter list names for each sample or treatment after “ANOVA(“ separate by commas
Close parenthesis and hit ENTER
Example: ANOVA(L1,L2,L3)

11. Summary and Homework Summary Homework
ANOVA is a method that tests whether three, or more, means are equal
One-Way ANOVA is applicable when there is only one factor that differentiates the groups
Not rejecting H0 means that there is not sufficient evidence to say that the group means are unequal
Rejecting H0 means that there is sufficient evidence to say that group means are unequal
Homework
pg ; 1-4, 6, 7, 11, 13, 14, 19

12. Problem 19 TI-83 Calculator Output
One-way ANOVA F= p=
Factor
df=2
SS=1.1675
MS=
Error
df=15
SS=
MS=
Sxp=