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The t-test

Inferences about Population Means

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Questions

How are the distributions ofz and trelated?

Given that

construct a rejection region. Draw a pictureto illustrate.

What is the standard error of the difference

between means? What are the factors that

influence its size?

01.2;49;14;75:;75: )48,05(.10 tNsHH y

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Questions (2)

What are the main uses of the t-test?

Give a concrete example of the use of the

{one sample, independent samples, dependent

samples} t-test. State why the particular testis the right one to choose.

What is the importance of variance accounted

for?

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Confidence intervals in z

For large samples (N>100) can use z.

Suppose

Then

If

M

Mest

yz

.

)(

N

N

yy

N

sest

y

M1

)(

.

2

200;5;10:;10:10

NsHH y

35.

14.14

5

200

5.

N

sest

y

M

05.96.183.2;83.235.

)1011(11

pzy

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The tDistributionWe use twhen the population variance is unknown (the

usual case) and sample size is small (N

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Degrees of Freedom

For the tdistribution, degrees of freedom are always asimple function of the sample size, e.g., (N-1).

One way of explaining dfis that if we know the total or

mean, and all but one score, the last (N-1) score is not free to

vary. It is fixed by the other scores. 4+3+2+X = 10. X=1.

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Confidence Intervals in t

With a small sample size, we compute the same numbers

as we did for z, but we compare them to the tdistribution

instead of thez distribution.

25;5;10:;10: 10 NsHH y

125

5.

N

sest

y

M1

1

)1011(11

ty

064.2)24,05(. t 1

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Review

How are the distributions ofz and trelated?

Given that

construct a rejection region. Draw a pictureto illustrate.

01.2;49;14;75:;75: )48,05(.10 tNsHH y

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Difference Between Means (1)

Most studies have at least 2 groups(e.g., M vs. F, Exp vs. Control)[1 v 2sample]

If we want to know diff in populationmeans, best guess is diff in samplemeans.

Unbiased:

Variance of the Difference:

Standard Error:

2

2

2

121)var(

MMyy

212121 )()()( yEyEyyE

2

2

2

1 MMdiff

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Difference Between Means (2)

We can estimate the standard error of

the difference between means.

For large samples, can use z

2

2

2

1... MMdiff estestest

diffest

yy

diffz 2121 )(

3;100;12

2;100;10

0:;0:

222

111

211210

SDNy

SDNy

HH

36.100

13

100

9

100

4. diffest

05.;56.5

36.

2

36.

0)1210(

pzdiff

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Independent Samples t (1)

Looks just like z:

df=N1-1+N2-1=N1+N2-2

If SDs are equal, estimate is:

diffestyy

difft 2121 )(

21

2

2

2

1

211

NNNNdiff

Pooled variance estimate is weighted average:

)]2/[(])1()1[( 212222112 NNsNsNestPooled Standard Error of the Difference (computed):

21

21

21

2

22

2

11

2

)1()1(.

NN

NN

NN

sNsN

est diff

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Independent Samples t(2)

21

21

21

2

22

2

11

2

)1()1(. NN

NN

NN

sNsNest diff

diffest

yy

difft 2121 )(

7;83.5;20

5;7;18

0:;0:

2222

1

2

11

211210

Nsy

Nsy

HH

47.135

12

275

)83.5(6)7(4.

diffest

..;36.147.1

2

47.1

0)2018(sntdiff

tcrit= t(.05,10)=2.23

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Assumptions

The t-test is based on assumptions of

normality and homogeneity of variance.

You can test for both these (make sure

you learn the SAS methods).

As long as the samples in each group

are large and nearly equal, the t-test is

robust, that is, still good, even thoassumptions are not met.

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Review

What is the standard error of the

difference between means? What are

the factors that influence its size?

What are the assumptions of the t-test?

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Strength of Association (1)

Scientific purpose is to predict orexplain variation.

Our variable Y has some variance that

we would like to account for. There arestatistical indexes of how well our IVaccounts for variance in the DV. Theseare measures of how strongly or closely

associated our IVs and DVs are. Variance accounted for:

2

2

21

2

2

|

2

2

4

)(

YY

XYY

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Strength of Association (2)

How much of variance in Y is

associated with the IV?2

2

21

2

2

|

2

2

4

)(

YY

XYY

6420-2-4

0.4

0.3

0.2

0.1

0.0

Compare the 1st (left-most) curve with the curve in the

middle and the one on the right.

In each case, how

much of the variance

in Y is associated

with the IV, groupmembership? More

in the second

comparison. As

mean diff gets big, so

does variance acct.

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Association & Significance

Power increaseswith association

(effect size) and

sample size.

Effect size:

Significance =

effect size X sample

size.

pyy /)( 21

21

2

21

11

)(

NN

yyt

p

Increasing sample size does not increase effect size

(strength of association). It decreases the standard

error so power is greater. Widely misunderstood.

N

yt

2

)(

pooledSD

XXd 21

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Estimating Power (1)

If the null is false, the statistic is nolonger distributed as t, but rather as

noncentral t. This makes power

computation difficult. Hays (p. 334) presents an alternative

method based on strength of

association, that is, on

2

2

21

2

2

|

2

2

4

)(

YY

XYY

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Estimating Power (2)

Based on Hayss method, we find:

35.22)25(.2

)75(.]58.228.1[2

gn

Suppose alpha is .01, power

desired is .90, and variance

accounted for is .25. What is

n per group? Its 24 (23?) per

group or 48 all together.

(Hays says add one moreperson for luck. its wise

28.1)90(.)1( zz

58.2)005(.)2/( zz

Same problem, but variance a/c is .10, need 68/group.

Same again, but .15, need 43 per group. What if alpha =

.05?

2

22

)2/()1(

2

)1(][

zzng

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Dependent t(1)

Observations come in pairs. Brother, sister, repeated measure.

),cov(2 212

2

2

1

2 yyMMdiff

Problem solved by finding diffs between pairs Di=yi1-yi2.

1

)(2

2

N

DDs

i

D

N

sest DMD .N

DD

i )(

MDest

DEDt

.

)( df=N(pairs)-1

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Dependent t (2)

Brother Sister

5 7

7 8

3 35y 6y

Diff2 1

1 0

0 1

1D

58.3/1. MD

est

72.158.

1

.

)(

MDest

DEDt

1

1

)(2

N

DDsD

2)( DD

df=2; n.s.

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Review

What are the main uses of the t-test?

Give a concrete example of the use of the

{one sample, independent samples, dependent

samples} t-test. State why the particular testis the right one to choose.

What is the importance of variance accounted

for?