1. Statistical Inference for Managers
One Way Analysis of variance (ANOVA) By
Imran Khan

2. One way ANOVA
Suppose we want to compare the means of k populations with same variance. The procedure for testing the equality of population means in this setup is called One Way ANOVA. H0: μ1=μ2=……=μk H1: μ1≠μ2≠…… ≠μk x̅i=∑Xij/ni Xij denotes jth observation in the ith population

3. One way ANOVA Overall mean of sample observations x̅= ∑∑Xij/n
Or x̅= ∑nix̅i /n
Two types of variability:
Variability about individual sample means within k-groups of observations or within-groups variability.
Between-groups variability

4. One way ANOVA- formulas!
SS1=∑(X1j-x̅1)² SS2= ∑(X2j-x̅2)² SSW= SS1+SS2+…+ SSW= ∑∑(Xij-x̅i)² For between-groups variability: (x̅1-x̅)², (x̅2-x̅)², (x̅3-x̅)² SSG=∑ni(x̅i-x̅)² SST= total sum of squares SST= ∑∑(Xij-x̅)² SST=SSW+SSG

5. Total Sum of Squares= Within group SS + Between groups SS One way ANOVA Example: A cars B cars C cars

6. Mean Squares
If null hypothesis that population means are same is true, SSW and SSG can be used as a basis for estimating population variance. MSW= SSW/n-k MSW= within groups mean squared MSG= SSG/ k-1 MSG= Between groups mean squared

7. Mean Squares
Greater the discrepancy between MSG and MSW, stronger would be our suspicion that H0 is not true. F= MSG/ MSW H0: μ1=μ2=……=μk Reject H0 if MSG/ MSW> Fk-1, n-k, α

8. One way ANOVA table
Source of variation S.S Degree of Mean F-ratio freedom squares Between- groups SSG k-1 MSG F= MSG Within- groups SSW n-k MSW MSW Total SST n-1

9. Example Question
An instructor has a class of 23 students. At the beginning of the semester, each student is randomly assigned to one of four Teaching Assistants- Smiley, Haydon, Alleline or Bland. The students are encouraged to meet with their assigned teaching assistant to discuss difficult course material. At the end of the semester, a common examination is administered. The scores obtained by students working with these teaching assistants are shown in the table:

10. Smiley Haydon Alleline Bland
Calculate the within-groups, between-groups and total sum of squares.
Complete the ANOVA table and test the null hypothesis of equality of pop. Mean scores for the Teaching Assistants.