Chapter 20

t- Test for Two Matched Samples

Vodka & WhiskeyThe researcher randomly selected 10 Cranberry vodka drinkers and 7 Boilermaker drinkers. The vodka group had a mean drunkenness of 52 and the whiskey group had a mean drunkenness of 39.

Previous Experiment Previous Decision

The drunkenness of vodka drinkers and whiskey drinkers is different

Question:Who typically drinks Cranberry Vodkas and who drinks Boilermakers?

Women Men2nd QuestionWho typically weighs more?

Lighter Heavier

3rd Question: Was the difference in the previous experiment due to different alcohol or different weights?

Two Independent Sample t-Test

Factors that affect the standard error (s 1 -2)Population Standard Deviation

Random Sampling and Sample Size

Variable estimates of the Variability between the selected samples

Possibility that the samples differ in the other factors that affect the measure.

– e.g. Mostly light group vs mostly heavy group

Always a problem with random selection

Two Independent Sample t-Test

Factors that affect the standard error (s 1 -2)Population Standard Deviation

Random Sampling and Sample Size

Variable estimates of the

Variability between the selected samples

Always going to affect the error

Can we minimize this factor by forcing the selected samples to be similar?

Matched Samples t-Test

Reducing a Two-Sample test into a Single Sample Comparison

(Single Sample t-Test)

Matched Samples t-Test FAQ

What does matching mean?An observation in one sample is paired with an observation in the other sample?

Matched Samples t-Test FAQ

How does matching happen?Pick a matching variable

Must be correlated to the dependent measure

e.g., weight and drunkenness

Randomly select 2 participants who are equal on the matching factor

Randomly assign 1 participant to each groupVodka: 110lb, 150lb, 190lb

Whiskey: 110lb, 150lb, 190lb

Matched Samples t-Test FAQ

What should matching do?Reduce the between sample variability

e.g., Affected by weight the same

Reduce the standard error.

What is special about the matched samples t-test?

Mathematically reduces the standard error.

Brings the critical scores closer to zero

Makes the test more sensitive.

Matched Samples t-Test FAQ

What does the test assume?Populations have a normal shape

…. or at least the sampling distributions are normal

Homogeneity of VariancePopulation have the same variability

Matching variable is correlated to the dependent measure

Matched Samples t-Test FAQ

What is a difference score? Actual dependent measure used by the test.

It is the difference between the paired scoresD110lb = Vodka110lb - Whiskey110lb

What is the null hypothesis?The groups are equal

Vodka110lb - Whiskey110lb = 0

D = 0

Matched Samples t-Test FAQ

Any other requirements?Equal sample sizes!!!!!

Hypothesis Test 11

Vodka & WhiskeyMatched Samples t-Test

Are all types of alcohol the same, even if the proofs are the same? This question was raised by a researcher who had observed vodka drinkers and noticed that they seemed to get drunk faster than whiskey drinkers. To test whether whiskey is the same or different than vodka, the researcher decided to compare people who drank 3 Cranberry Vodka’s to people to drank 3 Boilermakers. Since weight is known to affect drunkenness, the researcher has matched the samples to make sure that weight is evenly distributed between the group (on next slide).What will the researcher conclude at a .01 level of significance.

Vodka/Whiskey: Matched t-Test

Vodka/Whiskey: Matched t-TestStep 0) Convert to Difference Scores

Weight Vodka Whiskey

110 52 63

130 47 42

150 34 36

170 30 31

190 24 20

210 25 19 +6

+4

-1

-2

+ 5

- 11

D

Vodka/Whiskey: Matched t-TestStep 0) Convert to Difference Scores

+6

+4

-1

-2

+ 5

- 11

DD = D / ndif

= / 6= .16

(D2sD

=

D)2

ndif

ndif - 1

Vodka/Whiskey: Matched t-TestStep 0) Convert to Difference Scores

+6

+4

-1

-2

+ 5

- 11

D D = D / ndif

= / 6= .16

(D2sD

=

D)2

ndif

ndif - 1

sD

=

)2

66 - 136

16

1

4

25

121

D2

= 6.37

Vodka/Whiskey: Matched t-Test

Step 1) Rewrite the research question“Does the mean drunkenness of the vodka population equal the mean drunkenness of the whiskey population.”

Step 2) Write the statistical hypothesesH0: D= 0

H1: D 0

A)

Vodka/Whiskey: Matched t-Test=.16sD=6.37ndif = 6dfdif = 5 = .01

Is the mean of the vodka pop. the same as the whiskey pop?

HypothesisH0: D=0 H1: D 0

Step 3) Form Decision Rulea) Draw Normal Curveb) Shade in c) Mark Rejection Region(s)d) Determine Critical Scorese) Write conditions for rejection H0

C) Reject H0 Fail to reject H0 Reject H0

B).005 .005

t(5)crit= -4.032

t(5)crit= +4.032

Decision Rule: Reject H0

• tobt < -4.032• tobt > +4.032

E)

D

=.16sD=6.37ndif = 6dfdif = 5 = .01

Is the mean of the vodka pop. the same as the whiskey pop?

HypothesisH0: D=0 H1: D 0Decision Rule: Reject H0

• tobt < -4.032• tobt > +4.032

E)

Vodka/Whiskey: Matched t-Test

D = hyp

= 0D

=

6

=2.45

= 2.6

sDsD=ndif

=.16sD=6.37ndif = 6dfdif = 5 = .01

Is the mean of the vodka pop. the same as the whiskey pop?

HypothesisH0: D=0 H1: D 0Decision Rule: Reject H0

• tobt < -4.032• tobt > +4.032

E)

D

Vodka/Whiskey: Matched t-Test

Based upon a sampling distribution with & = 2.6

Step 4) Calculate Test Statistic

D = sD

tobt = D

sD

D -

tobt = .16 - 0

2.6

= .06

Vodka/Whiskey: Matched t-Test

Step 6) Interpret Decision

•Fail to Reject H0

Step 5) Make Decision

•We have no evidence to suggest that the drunkenness of vodka drinkers is different from whiskey drinkers.

=.16sD=6.37ndif = 6dfdif = 5 = .01

Is the mean of the vodka pop. the same as the whiskey pop?

HypothesisH0: D=0 H1: D 0Decision Rule: Reject H0

• tobt < -4.032• tobt > +4.032

E)

D

Based upon a sampling distribution with & = 2.6D = sD

tobt = .06

Why Matching Works!

Why can we assume the standard error is reduced (if the matching variable is correlated with the

dependent measure)

Why Matching Works

Weight & Drunkenness Correlation

20

30

40

50

60

70

100 110 120 130 140 150 160 170 180 190 200 210 220

Weight (lbs)

Dru

nken

ness

When Matching Doesn’t Work

IQ & Drunkenness Correlation

20

30

40

50

60

70

70 80 90 100 110 120 130

IQ

Dru

nken

ness

Matched Sample t-Test

Matched Sample t-Test and the correlation assumption.

The test does not know if the matching variable is correlated

Assumes it is correlated because you selected itDrops the standard error estimate

If the matching variable is not correlated…Standard error has not actually decreased

….. but the test lowered it anyway.

Repeated Measures

The Ultimate Matching

Repeated MeasuresRepeated Measures

Using the same participants in both conditions

Concern: Carry-over EffectsSpill-Over

Effects of drugs Linger

Practice effectsSecond time the participant has done the task

Counter-balancingSome participants given the conditions in reverse order

To match, or not to match…

That is the question……

Matched vs Independent TestsWhich test is more sensitive?

Independent Matched

Degrees of Freedom

More Less

Standard Error

Larger Smaller

Matched vs Independent TestsWhich test is more sensitive?

Independent Matched

Degrees of Freedom

Brings in Critical Values

Spreads out Critical Values

Standard Error

Spreads out Critical Values

Brings in Critical Values

… has a much

greater effect on critical scores?

Matched vs Independent TestsWhich test is more sensitive?

Independent Matched

Degrees of Freedom

Brings in a little

Spreads out a little

Standard Error

Spreads out a lot

Brings in a lot

… has a much

greater effect on critical scores?

MoreSensitive

Two Sample Tests

Final Notes

Two Sample Tests

One-Tailed TestsMake the population with the larger hypothesized mean population 1

Always an upper-critical test

Confidence IntervalsConfident that the difference between the population means is within the interval

…. not about the value of the population means