1. Comparing Three or More Means
Chapter 13
Comparing Three or More Means

2. Comparing Three or More Means (One-Way Analysis of Variance)
Section
13.1
Comparing Three or More Means (One-Way Analysis of Variance)

3. Analysis of Variance (ANOVA) is an inferential method used to test the equality of three or more population means.
CAUTION!
Do not test H0: μ1 = μ2 = μ3 by conducting three separate hypothesis tests, because the probability of making a Type I error will be much higher than α.

4. Requirements of a One-Way ANOVA Test
There must be k simple random samples; one from each of k populations or a randomized experiment with k treatments.
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 must have the same variance; that is, each treatment group has the population variance σ2.

5. Testing a Hypothesis Regarding k = 3
H1: At least one population mean is different from the others

6. Verifying the Requirement of Equal
Population Variance
The one-way ANOVA procedures may be used if the largest sample standard deviation is no more than twice the smallest sample standard deviation.

7. Parallel Example 1: Verifying the Requirements of ANOVA
The following data represent the weight (in grams) of pennies minted at the Denver mint in 1990,1995, and Verify that the requirements in order to perform a one-way ANOVA are satisfied.

9. Solution
The 3 samples are simple random samples.
The samples were obtained independently.
Normal probability plots for the 3 years follow. All of the plots are roughly linear so the normality assumption is satisfied.

10. Solution
4. The sample standard deviations are computed for each sample using Minitab and shown on the following slide. The largest standard deviation is not more than twice the smallest standard deviation (2• = > ) so the requirement of equal population variances is considered satisfied.

11. Descriptive Statistics
Variable N Mean Median TrMean StDev SE Mean
Variable Minimum Maximum Q Q3

12. The basic idea in one-way ANOVA is to determine if the sample data could come from populations with the same mean, μ, or suggests that at least one sample comes from a population whose mean is different from the others.
To make this decision, we compare the variability among the sample means to the variability within each sample.

13. We call the variability among the sample means the between-sample variability, and the variability of each sample the within-sample variability.
If the between-sample variability is large relative to the within-sample variability, we have evidence to suggest that the samples come from populations with different means.

14. ANOVA F-Test Statistic
The analysis of variance F-test statistic is given by

15. Computing the F-Test Statistic
Step 1: Compute the sample mean of the combined data set by adding up all the observations and dividing by the number of observations. Call this value .
Step 2: Find the sample mean for each sample (or treatment). Let represent the sample mean of sample 1, represent the sample mean of sample 2, and so on.
Step 3: Find the sample variance for each sample (or treatment). Let represent the sample variance for sample 1, represent the sample variance for sample 2, and so on.

16. Computing the F-Test Statistic
Step 4: Compute the sum of squares due to treatments, SST, and the sum of squares due to error, SSE.
Step 5: Divide each sum of squares by its corresponding degrees of freedom (k – 1 and n – k, respectively) to obtain the mean squares MST and MSE.
Step 6: Compute the F-test statistic:

17. Parallel Example 2: Computing the F-Test Statistic
Compute the F-test statistic for the penny data.

18. Solution Step 1: Compute the overall mean.
Step 3: Find the sample variance for each treatment (year).

19. Solution
Step 4: Compute the sum of squares due to treatment, SST, and the sum of squares due to error, SSE.
SST = 11( − )2 + 11( − )2 + 11(2.5 − )2 =
SSE = (11 − 1)(0.0005) + (11 − 1)(0.0006) + (11 − 1)(0.0002) =
Step 5: Compute the mean square due to treatment, MST, and the mean square due to error, MSE.

20. Solution
Step 6: Compute the F-statistic.
The results of the computations for the data that led to the F-test statistic are presented in the following table.
ANOVA Table:
Source of Variation
Sum of Squares
Degrees of Freedom
Mean Squares
F-Test Statistic
Treatment
0.0009 2
0.0005
1.25
Error
0.013 30
0.0004
Total
0.0139 32

21. Example Compute the F-test statistic for the following data. 𝒙 𝟏 𝒙 𝟐
𝒙 𝟑 4 7 10 5 8 6 9 11 13

22. Example
Suppose the teeth researcher wans to determine if there is a difference in the mean shear bond strength among the four treatment groups at the 𝛼=0.05 level of significance.

23. Example
The data in the table represent the number of pods on a random sample of soybean plants for various plot types. An agricultural researcher wants to determine if the mean numbers of pods for each plot type are equal. Use the 𝛼=0.05 level of significance.

24. Plot Type
Pods
Liberty 32 31 36 35 41 34 39 37 38
No till 30 27 40 33 42
Chisel plowed 24 23