Transcript Chapter 9.

Independent Samples
1. Random Selection:
Everyone from the Specified Population has an Equal Probability
Of being Selected for the study (Yeah Right!)
2. Random Assignment:
Every participant has an Equal Probability of being in the Treatment
Or Control Groups
The Null Hypothesis
•Both groups from Same Population
No Treatment Effect
•Both Sample Means estimate Same Population Mean
Difference in Sample Means reflect Errors of Estimation of Mu
X-Bar1 + e1 = Mu
X-Bar2 + e2 = Mu
(Mu – X-Bar1 = e1)
(Mu – X-Bar2 = e2)
Errors are Random and hence Unrelated
Expectation
If Both Samples were selected from the Same Population:
How much should the Sample Means Disagree about Mu?
X-Bar1 – X-Bar2
•Errors of Estimation decrease with N
•Errors of Estimation increase with Population Heterogeneity
The Expected Disagreement
The Standard Error of a Difference:
SEX-Bar1-X-Bar2
The Average Difference between two Sample Means
The Expected Difference between two Sample Means
•When they are Estimating the Same Mu
•68% chance of this much Or Less
•95% chance of (this much x 2) Or Less
Actually this much x 1.96, if you know sigma
Rounded up to 2
Expectation: The Standard Error
of the Difference
The Expected Disagreement between two Sample Means (if H0 true)
T for Treatment Group
C for Control Group
 SEM for
Treatment
Group
 SEM for
Control
Group
Add the Errors and take the Square Root
Evaluation
Compare the Difference you Got to the Difference you would Expect
If H0 true
What you Got
?
What you Expect
df = n1 + n2 - 2
Evaluation
Compare the Difference you Got to the Difference you would Expect
If H0 true
What you Got
a) If they agree: Keep H0
?
b) If they disagree: Reject H0
What you Expect
Is TOO DAMN
BIG!
Burn This!
Power
The ability to find a relationship when it exists
•Errors of Estimation and Standard Errors of the Difference
decrease with N
Use the Largest sample sizes possible
•Errors of Estimation increase with Population Heterogeneity
Run all your subjects under Identical Conditions
(Experimental Control)
Power
What if your data look like this?
Everybody increased their score (X-bar1 – X-Bar2),
but heterogeneity among subjects (SEM1 & SEM2) is large
40
30
20
Value
10
Pre-T est
0
Post-T est
1
2
Case Number
3
4
5
6
7
8
9
10
Power
Correlated Samples Designs:
•Natural Pairs: E.G.: Father vs. Son
Measuring liberal attitudes
•Matched Pairs: Matching pairs of students on I.Q.
One of each pair gets treatment (e.g., teaching with technology
•Repeated Measures:
Measure Same Subject Twice (e.g., Pre-, Post-therapy)
Look at differences between Pairs of Data Points, ignoring Between
Subject differences
Correlated Samples
Smaller denominator
Makes t bigger, hence
More Power
Same as 
usual
Minus strength 
of Correlation
If r=0, denominator is the same, but df is smaller
Effect Size
A weighted average of
Two estimates of Sigma
•What are the Two Ts of Research?
•What is better than computing Effect Size?
Confidence Interval
Use 2-tailed t-value at
95% confidence level
With N1 + N2 –2 df
 Best Estimate 
N-1 df
Does the Interval cross Zero?
76
74
72
Mean +- 2 SE HEIGHT
70
68
66
64
62
N=
SEX
20
10
f
m
Group Statistics
HEIGHT
SEX
f
m
N
20
10
Mean
64.9500
72.3000
Std. Deviation
2.45967
1.82878
Std. Error
Mean
.55000
.57831
Independent Samples Test
Levene's Test for
Equality of Variances
F
HEIGHT
Equal variances
ass umed
Equal variances
not as sumed
1.352
Sig.
.255
t-tes t for Equality of Means
t
df
Sig. (2-tailed)
Mean
Difference
Std. Error
Difference
95% Confidence
Interval of the
Difference
Lower
Upper
-8.338
28
.000
-7.3500
.88151
-9.15568
-5.54432
-9.210
23.527
.000
-7.3500
.79809
-8.99893
-5.70107
72
70
68
66
64
62
N=
HAIR
1
11
18
b
n
Independent Samples Test
Levene's Test for
Equality of Variances
F
HEIGHT
Equal variances
ass umed
Equal variances
not as sumed
.748
Sig.
.395
t-tes t for Equality of Means
t
df
Sig. (2-tailed)
Mean
Difference
Std. Error
Difference
95% Confidence
Interval of the
Difference
Lower
Upper
-1.527
27
.139
-2.4242
1.58807
-5.68268
.83420
-1.573
23.314
.129
-2.4242
1.54102
-5.60972
.76123
Assumptions of the t-Test
Both (if more than one) population(s):
1. Normally distributed
2. Equal variance
Violations of Assumptions:
Robust unless gross
Transform scores (e.g. take log of each score)
Power
Power = 1 – Beta
Theoretical (Beta usually unknown)
Reject H0:
Decision is clear, you have a relationship
Fail to reject H0:
Decision is unclear, you may have failed to find a Relationship
due to lack of Power
Power
1. Increases with Effect Size (Mu1 – Mu2)
2. Increases with Sample Size
If close to p<0.05 add N
3. Decreases with Standard Error of the Difference (denominator)
Minimize by
• Recording data correctly
• Use consistent criteria
• Maintain consistent experimental conditions (control)
• (Increasing N)