Chapter 12 ANOVA

ANOVA (3 or more means simultaneously)

• The hypotheses are:– Ho: μ1 = μ2 = μ3 = μ4– Ha: At least one mean is different

• We use an F-test. – d.f.N = (k – 1)– d.f.D = (N – k)

• Where k = # of groups and N = Total of all the groups.

ANOVA Assumptions

• There are three assumptions for the F Test comparing three or more means.1. The populations from which the samples were

obtained must be normally distributed.

2. The samples must be independent of one another.

3. The variances of the populations must be equal.

ANOVA

• With the F test, two different estimates of the population variance are made. 1. Between-group variance – the variance of the

means.

2. Within-group variance – the variance using all the data

• Not affected by differences in the means

ANOVA

• If there is NO difference in the means, the between-group variance estimate will be approximately equal to the within-group variance. This makes the F test stat. close to 1. This will result in a decision to not reject the null. Conversely F test stats bigger than one will indicate a difference in at least one of the means and the decision will be to reject the null.

Example – ANOVA HT• A researcher wishes to try three different

techniques to lower the blood pressure of individuals diagnosed with high blood pressure. The subjects are randomly assigned to three groups (medication, exercise, and diet). After four weeks, the reduction in each person’s blood pressure is recorded. At α = 0.05, test the claim that there is no difference among the means.

Example – ANOVA HT

• Step 1– Ho: μ1 = μ2 = μ3 = μ4– Ha: At least one mean is different

• Step 2– α = 0.05

• Step 3– F( 2,12)

Example – ANOVA HT

• Step 4– F( 2,12) = = 9.17

• Use your calculator to put the data into lists

• STAT -> TEST -> ANOVA(L1,L2,L3)

– P-value = 0.004

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ANOVA

• Step 5– 0.004 < 0.05– Reject Ho

• Step 6– There is sufficient evidence to suggest that at

least one mean is different.

Which mean is different?

• In order to figure out which mean is different from the others we would have to do a pair-wise comparison. There two tests that you can use to do this.

1.Tukey Test

2.Sheffé Test