Statistics Lecture 16

zLast Day: Two-Sample T-test (10.2 and 10.3) zToday: Comparison of Several Treatments ( )

zLast day, we looked at comparing means for two treatments zWhen more than two treatments are being compared, we will use a statistical technique call the Analysis of Variance (ANOVA) zThe same underlying assumptions apply in the ANOVA situation a the two independent samples case

Example (Pulp Mill) zAn important measure of performance at pulp mills is based on pulp brightness measured by a reflectance meter zAn investigation was performed (Sheldon, 1960; Industrial and Engineering Chemistry ) to investigate if there is a difference in product quality for different mill operators zWant to see if there are differences in the reflectance for different operators

zData:

ANOVA Situation zSituation: yHave k independent random samples yEach sample comes from a normal population yThe population standard deviations are equal yWant to test test a hypothesis about the equality of the population means

Structure of Data zHave k independent random samples from k populations…sample size from each pop. may be different zDenote j th observation from the i th population as y ij

Estimating zHave assumed that data from each population comes from independent normal distributions with equal standard deviations (variances) zThat is, has a distribution zIf we wanted to estimate based on the data from only 1 population, we would use zCombining the data from all of the populations:

Another estimate for zWhy are we doing this? zWhat is the null hypothesis we have in mind? zSuppose H 0 is true, how could we estimate the mean? zVariance about true mean:

zWhen the null hypothesis is true, we expect zWhen it is false zPotential Test Statistic

More Formal Approach zModel for comparing k treatments: y for i =1, 2, …, k and j =1, 2, …, n i ywhere is the i th population mean and ye ij had a distribution zWant to test: y

zSum of Squares for treatment zSum of Squares for Error (residual) zTotal Sum of Squares

zDegrees of freedom zMean Squares zTest Statistic

zHypotheses zP-value

zANOVA Table:

Example (Pulp Mill): zData:

zSummary Statistics:

Plot of Responses By Operator

ANOVA Table