Lesson 13 - 1

1. Lesson 13 - 1 Comparing Three or More Means ANOVA(One-Way Analysis of Variance)

2. Objectives • Verify the requirements to perform a one-way ANOVA • Test a claim regarding three or more means using one way ANOVA

3. Vocabulary • ANOVA – Analysis of Variance: inferential method that is used to test the equality of three or more population means • Robust – small departures from the requirement of normality will not significantly affect the results • Mean squares – is an average of the squared values (for example variance is a mean square) • MST – mean square due to the treatment • MSE – mean square due to error • F-statistic – ration of two mean squares

4. One-way ANOVA Test Requirements • There are k simple random samples from k populations • The k samples are independent of each other; that is, the subjects in one group cannot be related in any way to subjects in a second group • The populations are normally distributed • The populations have the same variance; that is, each treatment group has a population variance σ2

5. ANOVA Requirements Verification • ANOVA is robust, the accuracy of ANOVA is not affected if the populations are somewhat non- normal or do not quite have the same variances • Particularly if the sample sizes are roughly equal • Use normality plots • Verifying equal population variances requirement: • Largest sample standard deviation is no more than two times larger than the smallest

6. ANOVA – Analysis of Variance Computing the F-test Statistic 1. Compute the sample mean of the combined data set, x • Find the sample mean of each treatment (sample), xi • Find the sample variance of each treatment (sample), si2 • Compute the mean square due to treatment, MST • Compute the mean square due to error, MSE • Compute the F-test statistic: mean square due to treatment MST F = ------------------------------------- = ---------- mean square due to error MSE ni(xi – x)2 (ni – 1)si2 MST = -------------- MSE = ------------- k – l n – k k Σ k Σ n = 1 n = 1

7. MSE and MST • MSE -mean square due to error, measures how different the observations, within each sample, are from each other • It compares only observations within the same sample • Larger values correspond to more spread sample means • This mean square is approximately the same as the population variance • MST - mean square due to treatment, measures how different the samples are from each other • It compares the different sample means • Larger values correspond to more spread sample means • Under the null hypothesis, this mean square is approximately the same as the population variance

8. ANOVA – Analysis of Variance Table

9. Excel ANOVA Output • Classical Approach: • Test statistic > Critical value … reject the null hypothesis • P-value Approach: • P-value < α (0.05) … reject the null hypothesis

10. TI Instructions • Enter each population’s or treatments raw data into a list • Press STAT, highlight TESTS and select F: ANOVA( • Enter list names for each sample or treatment after “ANOVA(“ separate by commas • Close parenthesis and hit ENTER • Example: ANOVA(L1,L2,L3)

11. Summary and Homework • Summary • ANOVA is a method that tests whether three, or more, means are equal • One-Way ANOVA is applicable when there is only one factor that differentiates the groups • Not rejecting H0 means that there is not sufficient evidence to say that the group means are unequal • Rejecting H0 means that there is sufficient evidence to say that group means are unequal • Homework • pg 685-691; 1-4, 6, 7, 11, 13, 14, 19

12. Problem 19 TI-83 Calculator Output • One-way ANOVA • F=5.81095 • p=.013532 • Factor • df=2 • SS=1.1675 • MS=0.58375 • Error • df=15 • SS=1.50686 • MS=.100457 • Sxp=0.31695