Post on 31-Dec-2015
description
While you wait:
Enter the following in your calculator. Find the mean and sample variation of
each group.
Bluman, Chapter 12 1
Chapter 12
Analysis of Variance
McGraw-Hill, Bluman, 7th ed., Chapter 12 2
3
21Per 5-8; Wed Schedule Sec 12.1
23ACT testing; Classes don’t meet
25 Sec12.2
28 Sec 12.3
30 Sec 12.3 continued.
May 2 Various topics Chapter 13
5 Various topics Chapter 13
7 Various topics Chapter 13
9 Various topics Chapter 14
12 Various topics Chapter 14
14Test 12-14
16 Review for Final Exam
19 Review for Final Exam
2 Review for Final Exam 1
23 Final Exam
Chapter 12 Overview Introduction 12-1 One-Way Analysis of Variance
12-2 The Scheffé Test and the Tukey Test
12-3 Two-Way Analysis of Variance
Bluman, Chapter 12 4
Chapter 12 Objectives1. Use the one-way ANOVA technique to
determine if there is a significant difference among three or more means.
2. Determine which means differ, using the Scheffé or Tukey test if the null hypothesis is rejected in the ANOVA.
3. Use the two-way ANOVA technique to determine if there is a significant difference in the main effects or interaction.
Bluman, Chapter 12 5
What is the ANOVA Test?
• Remember the 2-Mean T-Test?• For example: A salesman in car sales wants
to find the difference between two types of cars in terms of mileage:
• Mid-Size Vehicles
• Sports Utility Vehicles
Car Salesman’s Sample
The salesman took an independent SRS from each population of vehicles:
Level n Mean StDev
Mid-size 28 27.101 mpg 2.629 mpg
SUV 26 20.423 mpg 2.914 mpg
If a 2-Mean TTest were done on this data:
T = 8.15 P-value = ~0
What if the salesman wanted to compare another type of car, Pickup Trucks in addition to the SUV’s and Mid-size vehicles?
Level n Mean StDev
Midsize 28 27.101 mpg 2.629 mpg
SUV 26 20.423 mpg 2.914 mpg
Pickup 8 23.125 mpg 2.588 mpg
This is an example of when we would use the ANOVA Test.
In a 2-Mean TTest, we see if the
difference between the 2 sample means is significant.
The ANOVA is used to compare multiple means, and see if the
difference between multiple sample means is significant.
Let’s Compare the Means…
Do these sample means look significantly different from each other?
Yes, we see that no two of these confidence intervals overlap, therefore the means are significantly different.
This is the question that the ANOVA test answers mathematically.
More Confidence Intervals
What if the confidence intervals were different? Would these confidence intervals be significantly different?
SignificantNot Significant
ANOVA Test Hypotheses
H0: µ1 = µ2 = µ3 (All of the means are equal)
HA: Not all of the means are equal
For Our Example:
H0: µMid-size = µSUV = µPickup
The mean mileages of Mid-size vehicles, Sports Utility
Vehicles, and Pickup trucks are all equal.
HA: Not all of the mean mileages of Mid-size vehicles,
Sports Utility Vehicles, and Pickup trucks are equal.
Introduction The F test, used to compare two variances,
can also be used to compare three of more means.
This technique is called analysis of variance or ANOVA.
For three groups, the F test can only show whether or not a difference exists among the three means, not where the difference lies.
Other statistical tests, Scheffé test and the Tukey test, are used to find where the difference exists.
Bluman, Chapter 12 13
12-1 One-Way Analysis of Variance When an F test is used to test a
hypothesis concerning the means of three or more populations, the technique is called analysis of variance (commonly abbreviated as ANOVA).
Although the t test is commonly used to compare two means, it should not be used to compare three or more.
Bluman, Chapter 12 14
Assumptions for the F TestThe following assumptions apply when using the F test to compare three or more means.
1. The populations from which the samples were obtained must be normally or approximately normally distributed.
2. The samples must be independent of each other.
3. The variances of the populations must be equal.
Bluman, Chapter 12 15
The F Test In the F test, two different estimates of
the population variance are made. The first estimate is called the between-
group variance, and it involves finding the variance of the means.
The second estimate, the within-group variance, is made by computing the variance using all the data and is not affected by differences in the means.
Bluman, Chapter 12 16
The F Test If there is no difference in the means, the
between-group variance will be approximately equal to the within-group variance, and the F test value will be close to 1—do not reject null hypothesis.
However, when the means differ significantly, the between-group variance will be much larger than the within-group variance; the F test will be significantly greater than 1—reject null hypothesis.
Bluman, Chapter 12 17
Chapter 12Analysis of Variance
Section 12-1Example 12-1
Page #630
Bluman, Chapter 12 18
Example 12-1: Lowering Blood PressureA researcher wishes to try three different techniques to lower the blood pressure of individuals diagnosed with high blood pressure. The subjects are randomly assigned to three groups; the first group takes medication, the second group exercises, and the third group follows a special diet. After four weeks, the reduction in each person’s blood pressure is recorded. At α = 0.05, test the claim that there is no difference among the means.
Bluman, Chapter 12 19
Example 12-1: Lowering Blood Pressure
Bluman, Chapter 12 20
Step 1: State the hypotheses and identify the claim.H0: μ1 = μ2 = μ3 (claim)H1: At least one mean is different from the others.
Example 12-1: Lowering Blood Pressure
Bluman, Chapter 12 21
Step 2: Find the critical value.Since k = 3, N = 15, and α = 0.05,d.f.N. = k – 1 = 3 – 1 = 2d.f.D. = N – k = 15 – 3 = 12The critical value is 3.89, obtained from Table H.
Example 12-1: Lowering Blood Pressure
Bluman, Chapter 12 22
Step 3: Compute the test value.a. Find the mean and variance of each sample
(these were provided with the data).
b. Find the grand mean, the mean of all values in the samples.
c. Find the between-group variance, .
10 12 9 4 1167.73
15 15
GM
XX
N
2Bs
2
2
1
i i GM
B
n X Xs
k
Example 12-1: Lowering Blood Pressure
Bluman, Chapter 12 23
Step 3: Compute the test value. (continued)c. Find the between-group variance, .
d. Find the within-group variance, .2Ws
22 1
1
i iB
i
n ss
n
4 5.7 4 10.2 4 10.3 104.808.73
4 4 4 12
2 2 2
2 5 11.8 7.73 5 3.8 7.73 5 7.6 7.73
3 1160.13
80.072
Bs
2Bs
Step 3: Compute the test value. (continued)
e. Compute the F value.
Step 4: Make the decision.
Reject the null hypothesis, since 9.17 > 3.89.
Step 5: Summarize the results.
There is enough evidence to reject the claim and conclude that at least one mean is different from the others.
Example 12-1: Lowering Blood Pressure
Bluman, Chapter 12 24
2
2 B
W
sF
s80.07
9.178.73
ANOVA Please see page 634 The between-group variance is sometimes
called the mean square, MSB.
The numerator of the formula to compute MSB is called the sum of squares between groups, SSB.
The within-group variance is sometimes called the mean square, MSW.
The numerator of the formula to compute MSW is called the sum of squares within groups, SSW.
Bluman, Chapter 12 25
Bluman, Chapter 12 26
Choice HIn ( ) enter the lists… follow each list by a comma
Bluman, Chapter 12 27
Scroll down to see the rest of the values.
ANOVA Summary Table: see page 633
Bluman, Chapter 12 28
Source Sum of Squares
d.f. MeanSquares
F
Between
Within (error)
SSB
SSW
k – 1
N – k
MSB
MSW
Total
MS
MSB
W
ANOVA Summary Table for Example 12-1, page 634
Compare the calculator results with the values on the chart.
29
Source Sum of Squares
d.f. MeanSquares
F
Between
Within (error)
160.13
104.80
2
12
80.07
8.73
9.17
Total 264.93 14
ANOVA Summary Table for Example 12-1, page 634
Bluman, Chapter 12 30
Source Sum of Squares
d.f. MeanSquares
F
Between
Within (error)
160.13
104.80
2
12
80.07
8.73
9.17
Total 264.93 14
ANOVA on your Calculator
Bluman, Chapter 12 31
Enter the data on the lists on your calculator.
Chapter 12Analysis of Variance
Section 12-1Example 12-2
Page #632
Bluman, Chapter 12 32
Bluman, Chapter 12 33
Full solution for ANOVA
For ANOVA testing in addition to performing hypothesis testing, an ANOVA summary table with proper values a and symbols should be included.
For examples of ANOVA summary table see page 634 and 635 of your text.
Bluman, Chapter 12 34
ANOVA Summary Table
Bluman, Chapter 12 35
Source Sum of Squares
d.f. MeanSquares
F
Between
Within (error)
SSB
SSW
k – 1
N – k
MSB
MSW
Total
MS
MSB
W
Use the values displayed by your calculator to determine the value of each of the following.
Example 12-2: Toll Road EmployeesA state employee wishes to see if there is a significant difference in the number of employees at the interchanges of three state toll roads. The data are shown. At α = 0.05, can it be concluded that there is a significant difference in the average number of employees at each interchange?
Bluman, Chapter 12 36
Example 12-2: Toll Road Employees
Bluman, Chapter 12 37
Step 1: State the hypotheses and identify the claim.H0: μ1 = μ2 = μ3 H1: At least one mean is different from the others
(claim).
Example 12-2: Toll Road Employees
Bluman, Chapter 12 38
Step 2: Find the critical value.Since k = 3, N = 18, and α = 0.05,d.f.N. = 2, d.f.D. = 15The critical value is 3.68, obtained from Table H.
Example 12-2: Toll Road Employees
Bluman, Chapter 12 39
Step 3: Compute the test value.a. Find the mean and variance of each sample
(these were provided with the data).
b. Find the grand mean, the mean of all values in the samples.
c. Find the between-group variance, .
7 14 32 11 1528.4
15 18
GM
XX
N
2Bs
2
2
1
i i GM
B
n X Xs
k
Example 12-2: Toll Road Employees
Bluman, Chapter 12 40
Step 3: Compute the test value. (continued)c. Find the between-group variance, .
d. Find the within-group variance, .2Ws
22 1
1
i iB
i
n ss
n
5 81.9 5 25.6 5 29.0 682.545.5
4 4 4 15
2 2 2
2 6 15.5 8.4 6 4 8.4 6 5.8 8.4
3 1459.18
229.592
Bs
2Ws
Step 3: Compute the test value. (continued)
e. Compute the F value.
Step 4: Make the decision.
Reject the null hypothesis, since 5.05 > 3.68.
Step 5: Summarize the results.
There is enough evidence to support the claim that there is a difference among the means.
Example 12-2: Toll Road Employees
Bluman, Chapter 12 41
2
2 B
W
sF
s229.59
5.0545.5
ANOVA Summary Table for Example 12-2
Bluman, Chapter 12 42
Source Sum of Squares
d.f. MeanSquares
F
Between
Within (error)
459.18
682.5
2
15
229.59
45.5
5.05
Total 1141.68 17
Homework
Read section 12.1 and take notes
Sec 12.1 page 637 #1-7 and 9, 11, 13
Bluman, Chapter 12 43
Thanks For Watching
A special thanks to Mr. Coons for all the help and advice.