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我拷,这回香港六合彩的叛逆可作得比我要彻底多了,香港六合彩如今已经完全是个叛徒了——香港六合彩找了个外星人作男朋友,香港六合彩根本就叛变了地球!!
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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