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

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

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

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

Total Sum of Squares= Within group SS + Between groups SS One way ANOVA Example: A cars B cars C cars 22.2 24.6 22.7 19.9 23.1 21.9 20.3 22.0 23.2 21.4 23.5 24.1 21.2 23.6 22.2 21.0 22.1 23.4 20.3 23.5

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

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

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

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:

Smiley Haydon Alleline Bland 78 80 79 69 93 68 70 84 79 59 61 76 97 75 74 88 82 85 81 68 63 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.