AP Statistics: ANOVA Section 2

In the previous section, we saw how to use ANOVA to test for a difference in means among several groups. However, that test only tells us

when differences exist, not which specific groups differ. The goal of this section is to adapt the inference procedures of section 13.1 to use

the results of the ANOVA analysis. This will allow us to find a confidence interval for the mean of

any group, find a confidence interval for a difference in means between two groups and

test when that difference is significant.

In earlier inference work, we used formulas such as those below for doing inference about a

single mean or a difference in means.

If we have found an ANOVA table based on samples from several groups, we make a couple of small adjustments to these formulas.* Estimate any standard deviation with from the ANOVA table. Since one of the conditions for the ANOVA is that the standard deviation is the same for each group, using gives an estimate that is based on all of the samples and not just one.* Use the error degrees of freedom, n – k, for any t- distributions. We often call the pooled standard deviation

Inference for Means After ANOVAAfter doing an ANOVA for a difference in means among k groups based on sample of size

Confidence interval for : Confidence interval for :

i

in

MSEtx *

jiji nn

MSEtxx11

)( *

If the ANOVA indicates that there are differences among the means: Pairwise test of vs :

where MSE is the mean square error from the ANOVA table and the t-distribution use n – k degrees of freedom

i j

ji

ji

nnMSE

xxt

11

Example: Let’s go back to the ants and sandwich fillings example from section 1.

Find and interpret a 95% confidence interval for the mean number of ants attracted to a peanut butter sandwich.

8

7.138080.234 )66.42,34.25(

42.66 and 25.34between issandwich butter peanut a to

attracted ants ofnumber mean that theconfident 95% are We

Find and interpret a 95% confidence interval for the difference in average ant counts between vegemite

and ham & pickles sandwiches.

8

1

8

17.138080.225.4975.30

25.6,75.30

6.25.- and 30.75-between is pickles &ham vs vegemite toattacted ants

ofnumber mean ebetween th difference theconfident 95% are We

Note: Since our confidence interval contains only negative numbers, this

implies that our two population means differ – this is not surprising since

ANOVA done earlier indicated that at least two of the groups have different

means.

Test at the 5% level for a difference in mean number of ants between vegemite and peanut

butter sandwiches.

PBVH :0

PBVaH : fillingsandwich that toattracted ants

ofnumber mean theis each where

1section ANOVA in checked wereconditions

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