STAT 224-3 Discussion # 04/30/07 TA: Quoc Tran
Single Factor ANOVA
•One Way ANOVA focuses on a comparison of more than two population
or treatment means.
•Null and alternative hypothesis
–H0:µ1=µ2=··· =µI
–Ha: At least two of the µi’s are different
where Iis the number of populations or treatments being compared, J
number of observations in each population or treatment, ntotal observa-
tions , n=I∗J.
•Basic Assumptions : The Ipopulation or treatment distributions are all
normal with the same σ2.
•Goal : To separate signal from noise
–Total sum of square: S ST o =PI
i=1 PJ
j=1(yij −¯y..)2
–Sum of square due to treatments (signal) (between) SST =PI
i=1 J∗
(¯yi. −¯y..)2
–Sum square of errors (noise) (within) SSE =PI
i=1 PJ
j=1(yij −¯yi.)2
–SS T o =SST +SSE
•ANOVA table ( Jobs. in each sample)
Source df Sum of squares Mean square F
Treatments I−1S ST MST =S ST /(I−1) MS T
MS E
Error I(J−1) SS E MSE =S SE /I (J−1)
Total n−1SST o
where
–Test statistic F−value =MS T
MS E has an Fdistribution with ν1=I−1
and ν2=I(J−1) when H0is true and the basic assumptions are
satisfied.
–P-value : P(Fν1,ν2> F −value). Reject H0when P-value ≤αor
F-value >critical value.
EXAMPLES
•Example 1
In an experiment to investigate the performance of four different brands
of spark plugs intended for use on a 125-cc two-stroke motorcycle, five
plugs of each brand were tested and the number of miles (at a constant
Email: tran@stat.wisc.edu 1 RM B248D, MSC
