Post on 30-Mar-2015
ANOVA: A Test of Analysis of Variance
By Harry Lee and Manik Kuchroo
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.
F Statistic
• Like any other test, the ANOVA test has its own test statistic
• The statistic for ANOVA is called the F statistic, which we get from the F Test
• The F statistic takes into consideration: – number of samples taken (I)– sample size of each sample (n1, n2, …, nI)– means of the samples ( 1, 2, …, I)– standard deviations of each sample (s1, s2,
…, sI)
x x x
Explaining the F-Statistic
• The F statistic determines if the variation between sample means is significant
This is what we are doing when we look at the 95% confidence intervals.
SampleEach In sIndividual AmongVariation
Means Sample AmongVariation
Another Look at the CI’s
From this picture, we can see that the variation between sample means is greater than the variation in each sample; therefore, F is large.
F Statistic Equation
INsnsnsn
Ixxnxxnxxn
FII
II
2222
211
2222
211
)1(...)1()1(1
)(...)()(
Rewritten as a formula, the F Statistic looks like this:
Weighing
Weighing
Standard Deviations (Squared)
Means (Squared)
The F Statistic
Degrees of Freedom
• The ANOVA test has 2 degrees of freedom:– N-I (Total number sampled – Number of Groups)
– I-1 (Number of Groups – 1)
• Some sample distributions with different degrees of freedom:
How About Our Example:
Data:
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
F value = 40.05
P-value = ~0 (Found from a table or using the Fcdf calculator command).
Conditions
As useful as the ANOVA test is, we can only use it if a number of conditions are met:
• We must take an independent SRS from each population that we sample
• All populations have the same standard deviation. (No population’s standard deviation is double another’s)
• All of the populations must be normally distributed
Testing the Conditions
• The salesman had originally taken independent SRS’s.
• The second condition is fulfilled since no sample has more than twice the standard deviation of any other.
• To test the third condition, whether the populations being sampled are normally shaped, we must look at the histograms of each sample:
Sample Histograms
2
4
6
8
10
12
14
16
Midsize16 18 20 22 24 26 28 30 32 34 36
Collection 1 Histogram
2
4
6
8
10
12
14
16
SUV16 18 20 22 24 26 28 30 32 34 36
Collection 1 Histogram
2
4
6
8
10
12
14
16
Pickup16 18 20 22 24 26 28 30 32 34 36
Collection 1 Histogram
All of the histograms appear to be relatively normally shaped.
Try a Problem
• Researchers are trying to see if the English AP scores from four different Massachusetts private schools are different. From each school, a random sample of students in the past year was taken and compared. Here are the results from the samples:
Results
School n Mean StDev
BB&N 23 4.3 0.4
Roxbury Latin 25 3.9 0.6
Winsor 26 4.2 0.3
Belmont Hill 29 3.1 0.3
Is there any significant difference between these schools’ AP English scores? (Assume that the populations are normally distributed)
Hypotheses
• H0: = µBB&N µRL = µWinsor = µBelHill
The mean AP English Test scores in BB&N, Roxbury Latin, Winsor, and Belmont Hill are all the same.
• HA: The mean AP English Test scores in BB&N, Roxbury Latin, Winsor, and Belmont Hill are not all the same.
Conditions
• Random samples taken
• All of the standard deviations are the same– No standard deviation is more than twice any
other.
• All of the populations are normally distributed
Doing out the F Statistic
F Curve
• Plug the F statistic into the F distribution (df = 3, 99). The shaded area has a p-value of nearly 0.
Interpretation
Since all the conditions were met, we have conclusive evidence (df = 3,99, p = 0) to reject the null hypothesis that the mean AP English Test scores in BB&N, Roxbury Latin, Winsor, and Belmont Hill are all the same.
Thanks For Watching
• A special thanks to Mr. Coons for all the help and advice.