Statistical Inference for Managers

One Way Analysis of variance (ANOVA)By

Imran Khan

One way ANOVASuppose 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=……=μkH1: μ1≠μ2≠…… ≠μk

x̅i=∑Xij/niXij denotes jth observation in the ith population

One way ANOVA

Overall mean of sample observations x̅= ∑∑Xij/n

Or x̅= ∑nix̅i /n

Two types of variability:1. Variability about individual sample means within

k-groups of observations or within-groups variability.

2. Between-groups variability

One way ANOVA- formulas!SS1=∑(X1j-x̅1)²SS2= ∑(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)²SST= total sum of squares

SST= ∑∑(Xij-x̅)²SST=SSW+SSG

Total Sum of Squares= Within group SS + Between groups SSOne way ANOVA Example:A cars B cars C cars22.2 24.6 22.719.9 23.1 21.920.3 22.0 23.221.4 23.5 24.121.2 23.6 22.221.0 22.1 23.420.3 23.5

Mean SquaresIf 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-kMSW= within groups mean squared

MSG= SSG/ k-1MSG= Between groups mean squared

Mean Squares

Greater the discrepancy between MSG and MSW, stronger would be our suspicion that H0 is not true.

F= MSG/ MSWH0: μ1=μ2=……=μkReject H0 if MSG/ MSW> Fk-1, n-k, α

One way ANOVA table

Source of variation S.S Degree of Mean F-ratio freedom squares

Between- groups SSG k-1 MSG F= MSGWithin- groups SSW n-k MSW MSWTotal SST n-1

Example QuestionAn 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:

Smiley Haydon Alleline Bland72 78 80 7969 93 68 7084 79 59 6176 97 75 7473 88 82 85 81 68 63

a) Calculate the within-groups, between-groups and total sum of squares.

b) Complete the ANOVA table and test the null hypothesis of equality of pop. Mean scores for the Teaching Assistants.