1. Statistics Are Fun! Analysis of Variance
Chapter 12
Analysis of Variance
Statistics Is Fun!

2. Goals List the characteristics of the F distribution
Statistics Are Fun!
Goals
List the characteristics of the F distribution
Conduct a test of hypothesis to determine whether the variances of two populations are equal
Discuss the general idea of analysis of variance
Organize data into a ANOVA table
Conduct a test of hypothesis among three or more treatment means
Statistics Is Fun!

3. Statistics Are Fun!
F Distribution
Used to test whether two samples are from populations having equal variances
Applied when we want to compare several population means simultaneously to determine if they came from equal population
ANOVA
Analysis of variance
In both situations:
Populations must be Normally distributed
Data must be Interval-scale or higher
Statistics Is Fun!

4. Characteristics Of The F Distribution
Statistics Are Fun!
Characteristics Of The F Distribution
Family of F Distributions
Each is determined by:
df in numerator
comes from pop. 1 which has larger sample variation
df in denominator
comes from pop. 2 which has smaller sample variation
F distribution is continuous
F Value can assume an infinite number of values from 0 to ∞
Value for F Distribution cannot be negative
Smallest value = 0
Positively skewed
Long tail is always to right
As # of df increases in both the numerator and the denominator, the distribution approaches normal
Asymptotic
As X increases the F curve approaches the X-axis
Statistics Is Fun!

5. Statistics Are Fun!
Why Do We Want To Compare To See If Two Population Have Equal Variances?
What if two machines are making the same part for an airplane?
Do we want the parts to be identical or nearly identical? Yes!
We would test to see if the means are the same: Chapter 10 & 11
We would test to see if the variation is the same for the two machines: Chapter 12
What if two stocks have similar mean returns?
Would we like to test and see if one stock has more variation than the other?
Statistics Is Fun!

6. Remember Chapter 11: Assumptions for small sample tests of means:
Statistics Are Fun!
Why Do We Want To Compare To See If Two Population Have Equal Variances?
Remember Chapter 11: Assumptions for small sample tests of means:
Sample populations must follow the normal distribution
Two samples must be from independent (unrelated) populations
The variances & standard deviations of the two populations are equal
Statistics Is Fun!

7. larger sample variance
Statistics Are Fun!
Conduct A Test Of Hypothesis To Determine Whether The Variances Of Two Populations Are Equal
To conduct a test:
Conduct two random samples
List population 1 as the sample with the largest variance:
n1 = # of observations
s1^2 = sample variance
n1 – 1 = df1 = degree of freedom (numerator for critical value lookup)
List population 2 as the sample with the smaller variance:
n2 = # of observations
s2^2 = sample variance
n2 – 1 = df2 = degree of freedom (denominator for critical value lookup)
Always list the sample
with the
larger sample variance
as population 1
(allows us to
use fewer
tables)
Statistics Is Fun!

8. Step 1: State null and alternate hypotheses
Statistics Are Fun!
Step 1: State null and alternate hypotheses
List the population with the suspected largest variance as population 1
Because we want to limit the number of F tables we need to use to look up values, we always put the larger variance in the numerator and the smaller variance in the denominator
This will force the F value to be at least 1
We will only use the right tail of the F distribution
Examples of Step 1:
Statistics Is Fun!

9. Step 2: Select a level of significance:
Statistics Are Fun!
Step 2: Select a level of significance:
Appendix G only lists significance levels: .05 and .01
Significance level = .10
.10/2 = .05
Use .05 table in
Appendix G
Significance level = .05
Use .05 table in
Appendix G
Statistics Is Fun!

10. Statistics Are Fun!
Step 3: Identify the test statistic (F), find critical value and draw picture
Look up Critical value in Appendix G and draw your picture
If you have a df
that is not listed
in the border,
calculate your F
by estimating
a value
between
two values.
HW #5:
df = 11,
use value
Between
10 & 12
Book says:
( )/2 =
3.105  3.10
Statistics Is Fun!

11. Step 4: Formulate a decision rule:
Statistics Are Fun!
Step 4
Step 4: Formulate a decision rule:
Example:
If our calculated test statistic is greater than 3.87, reject Ho and accept H1, otherwise fail to reject Ho
Statistics Is Fun!

12. Larger variance in numerator,
Statistics Are Fun!
Step 5
Step 5: Take a random sample, compute the test statistic, compare it to critical value, and make decision to reject or not reject null and hypotheses
Test Statistic F:
Example Conclusion for a two tail test:
Fail to reject null
“The evidence suggests that there is not a difference in variation”
Reject null and accept alternate
“The evidence suggests that there is a difference in variation”
Larger variance in numerator,
always!!
Let’s Look at
Handout 
Statistics Is Fun!

13. Statistics Are Fun!
Colin, a stockbroker at Critical Securities, reported that the mean rate of return on a sample of 10 software stocks was 12.6 percent with a standard deviation of 3.9 percent.
The mean rate of return on a sample of 8 utility stocks was 10.9 percent with a standard deviation of 3.5 percent. At the .05 significance level, can Colin conclude that there is more variation in the software stocks?
Example 1
Statistics Is Fun!

14. Step 3: The test statistic is the F distribution.
Statistics Are Fun!
Step 1: The hypotheses are
Step 2: The significance level is .05.
Step 3: The test statistic is the F distribution.
Example 1 continued
Statistics Is Fun!

15. Step 5: The value of F is computed as follows.
Statistics Are Fun!
Step 4: H0 is rejected if F>3.68 or if p < The degrees of freedom are n1-1 or 9 in the numerator and n1-1 or 7 in the denominator.
Step 5: The value of F is computed as follows.
H0 is not rejected. There is insufficient evidence to show more variation in the software stocks.
Example 1 continued
Statistics Is Fun!

16. ANOVA Analysis Of Variance
Statistics Are Fun!
ANOVA Analysis Of Variance
Technique in which we compare three or more population means to determine whether they could be equal
Assumptions necessary:
Populations follow the normal distribution
Populations have equal standard deviations ()
Populations are independent
Why ANOVA?
Using t-distribution leads to build up of type 1 error
“Treatment” = different populations being examined
Statistics Is Fun!

17. Case Where Treatment Means Are Different
Statistics Are Fun!
Case Where Treatment Means Are Different
Statistics Is Fun!

18. Case Where Treatment Means Are The Same
Statistics Are Fun!
Case Where Treatment Means Are The Same
Statistics Is Fun!

19. Example Of ANOVA Test To See If Four Treatment Means Are The Equal
Statistics Are Fun!
Example Of ANOVA Test To See If Four Treatment Means Are The Equal
22 students earned the following grades in Professor Rad’s class. The grades are listed under the classification the student gave to the instructor
Is there a difference in the mean score of the students in each of the four categories?
Use significance level α = .01
Statistics Is Fun!

20. Conduct A Test Of Hypothesis Among Four Treatment Means
Statistics Are Fun!
Conduct A Test Of Hypothesis Among Four Treatment Means
Step 1: State H0 and H1
H0 : µ1 = µ2 = µ3 = µ4
H1 : The Mean scores are not all equal (at least one treatment mean is different)
Step 2: Significance Level?
α = .01
Statistics Is Fun!

21. Step 3: Determine Test Statistic And Select Critical Value
Statistics Are Fun!
Step 3: Determine Test Statistic And Select Critical Value
Statistics Is Fun!

22. Step 4: State Decision Rule
Statistics Are Fun!
Step 4: State Decision Rule
If our calculated test statistic is greater than we reject H0 and accept H1, otherwise we fail to reject H0
Now we move on to Step 5: Select the sample, perform calculations, and make a decision…
Are you ready for a lot of procedures?!!
Statistics Is Fun!

23. The F critical value will determined whether we are close to 1 or not
Statistics Are Fun!
The idea is: If we estimate variation in two ways and use one estimate in the numerator and the other estimate in the denominator:
If we divide and get 1 or close to 1, the sample means are assumed to be the same
If we get a number far from 1, we say that the means are assumed to be different
The F critical value will determined whether we are close to 1 or not
Statistics Is Fun!

24. Let’s go calculate this!
Statistics Are Fun!
ANOVA Table So Far
Let’s go calculate this!
Statistics Is Fun!

25. Calculation 1: Treatment Means and Overall Mean
Statistics Are Fun!
Calculation 1: Treatment Means and Overall Mean
Statistics Is Fun!

26. Calculation 2: Total Variation
Statistics Are Fun!
Calculation 2: Total Variation
Statistics Is Fun!

27. Calculation 2: Total Variation
Statistics Are Fun!
Calculation 2: Total Variation
Statistics Is Fun!

28. Let’s go calculate this!
Statistics Are Fun!
ANOVA Table So Far
Let’s go calculate this!
Statistics Is Fun!

29. Calculation 3: Random Variation
Statistics Are Fun!
Calculation 3: Random Variation
Statistics Is Fun!

30. Calculation 3: Random Variation
Statistics Are Fun!
Calculation 3: Random Variation
Statistics Is Fun!

31. Let’s go calculate this!
Statistics Are Fun!
ANOVA Table So Far
Let’s go calculate this!
Statistics Is Fun!

32. Calculation 4: Treatment Variation
Statistics Are Fun!
Calculation 4: Treatment Variation
Simple Subtraction!
Statistics Is Fun!

33. Calculation 5: Mean Square (Estimate of Variation)
Statistics Are Fun!
Calculation 5: Mean Square (Estimate of Variation)
Statistics Is Fun!

34. Statistics Are Fun!
Calculation 6: F
Statistics Is Fun!

35. Statistics Are Fun!
Step 5: Make A Decision
Because is less than 5.09, we fail to reject H0
The evidence suggests that the mean score of the students in each of the four categories are equal (no difference)
Statistics Is Fun!

36. Summarize Chapter 12 List the characteristics of the F distribution
Statistics Are Fun!
Summarize Chapter 12
List the characteristics of the F distribution
Conduct a test of hypothesis to determine whether the variances of two populations are equal
Discuss the general idea of analysis of variance
Organize data into a ANOVA table
Conduct a test of hypothesis among three or more treatment means
Statistics Is Fun!