1

Tests with two+ groups

We have examined tests of means for a single group, and for a difference if we have a matched sample (as in husbands and wives)

Now we consider differences of means between two or more groups

2

Two sample t test

Compare means on a variable for two different groups. Income differences between males and

femalesAverage SAT score for blacks and whitesMean time to failure for parts manufactured

using two different processes

3

New Test - Same Logic

Find the probability that the observed difference could be due to chance factors in taking the random sample.If probability is very low, then conclude that difference did not happen by chance (reject null hypothesis)If probability not low, cannot reject null hypothesis (no diff. between groups)

4

Sampling Distributions

Mean1

Mean2

Note in this caseeach mean is notin the criticalregion of othersampling dist.

5

Sampling Distributions

Mean1

Mean2

Note each meanis well into thecritical region ofother samplingdistribution.

6

Sampling Dist. of Difference

Differenceof Means

HypothesizeZero Diff.

Big Differences

7

ProcedureCalculate means for each group

Calculate difference

Calculate standard error of difference

Test to see if difference is bigger than “t” standard errors (small samples)z standard errors (large samples)

t and z are taken from tables at 95 or 99 percent confidence level.

8

Standard error of difference

2121

222

211 11

2

)1()1(21 nnnn

snsns yy

Pooled estimate of standard deviation

Divide bysamplesizes

9

t test

21

21

yys

yyt

Difference of Means

Standard error ofdifference of means

If t is greater than table value of t for 95%confidence level, reject null hypothesis

10

Three or more groups

If there are three or more groups, we cannot take a single difference, so we need a new test for differences among several means.This test is called ANOVA for ANalysis Of VArianceIt can also be used if there are only two groups

11

Analysis of Variance

Note the name of the test says that we are looking at variance or variability.The logic is to compare variability between groups (differences among the means) and variability within the group (variability of scores around the mean)These are call the between variance and the within variance, respectively

12

The logic

If the between variance is large relative to the within variance, we conclude that there are significant differences among the means.

If the between variance is not so large, we accept the null hypothesis

13

Examples

Large Between

Small Between

Both exampleshave sameWithin

14

Variance

Calculate sum of squares and then divide by degrees of freedom

Three ways to do this

1

)( 2

n

YY

15

Total, Within, and BetweenTotal variance is the mean squared deviation of individual scores around the overall (total) mean

Within variance is the mean squared deviation of individual scores around each of the group means

Between variance is the mean squared deviation of group means around the overall (total) mean

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Total, Within, and Between

1

)( 2

ndf

yySS

T

T

Kndf

yySS

W

kW

2)(

1

)( 2

Kdf

yySS

B

kB

Total = SST/dfT

Within = SSW/dfW

Between = SSB/dfB

17

F test for ANOVA

The F statistic has a distribution somewhat like the chi-square. It made of the ratio of two variances.

For our purpose, we will compare the between and within estimates of variance

Create a ratio of the two -- called an F ratio. Between variance divided by the within variance

18

F-ratio

Table in the back of the book has critical values of the F statistic. Like the t distribution, we have to know degrees of freedom

Different than the t distribution, there are two different degrees of freedom we need

Between (numerator) and within (denominator)

19

Decision

If F-ratio for our sample is larger than the critical value, we reject the null hypothesis of no differences among the means

If F-ratio is not so large, we accept null hypothesis of no differences among the means

20

Example (three groups)

1 2 3 4 5 6 7 8 9 Observations

Overall mean is 5

222

222

222

)59()58()57(

)56()55()54(

)53()52()51(

TSS

60

21

Example (within)

1 2 3 4 5 6 7 8 9 Observations

222

222

222

)89()88()87(

)56()55()54(

)23()22()21(

WSS

2 5 8 Group Means

6

22

Example (between)

222

222

222

)58()58()58(

)55()55()55(

)52()52()52(

BSS

1 2 3 4 5 6 7 8 9 Observations2 5 8 Group MeansOverall mean is 5

54

23

F-ratio

Between variance divided by within variance.Between= 54 / 2 = 27 (remember k-1 degrees

of freedom, so df = 3-1Within = 6 / 6 = 1 (remember n-k degrees of

freedom, so df = 9-3

F-ratio is 27/1 with 2 and 6 df

Critical value (95%) of F is 5.14