1. SMOKERS NONSMOKERS Sample 1, size n1 Sample 2, size n2
Imagine the following “observational” study…
X = Survival Time (“Time to Death”) in two independent normally-distributed populations
SMOKERS
NONSMOKERS
X1 ~ N(μ1, σ1) 1 σ1
X2 ~ N(μ2, σ2)
Null Hypothesis
H0: μ1 = μ2, i.e.,
μ1 – μ2 = 0
(“No mean difference")
Test at signif level α 2 σ2
“The reason for the significance was that the smokers started out older than the nonsmokers.”
Sample 1, size n1
Sample 2, size n2
Suppose a statistically significant difference exists, with evidence that μ1 < μ2.
How do we prevent this criticism???

2.         POPULATION 1 POPULATION 2 X ~ N(μ1, σ1) 1 σ1
Now consider two dependent (“matched,” “paired”) populations…
X and Y normally distributed.
POPULATION 1
POPULATION 2
X ~ N(μ1, σ1) 1 σ1
Y ~ N(μ2, σ2)
Null Hypothesis
H0: μ1 = μ2, i.e.,
μ1 – μ2 = 0
(“No mean difference")
Test at signif level α 2 σ2
Classic Examples: Twin studies, Left vs. Right, Pre-Tx (Baseline) vs. Post-Tx, etc.
Common in human trials to match on Age, Sex, Race,…        
By design, every individual in Sample 1 is “paired” or “matched” with an individual in Sample 2, on potential confounding variables.
… etc….
… etc….

3.     POPULATION 1 POPULATION 2 X ~ N(μ1, σ1) 1 σ1 Y ~ N(μ2, σ2)
Now consider two dependent (“matched,” “paired”) populations…
X, and Y normally distributed.
POPULATION 1
POPULATION 2
X ~ N(μ1, σ1) 1 σ1
Y ~ N(μ2, σ2)
Null Hypothesis
H0: μ1 = μ2, i.e.,
μ1 – μ2 = 0
(“No mean difference")
Test at signif level α 2 σ2
D =
Classic Examples: Twin studies, Left vs. Right, Pre-Tx (Baseline) vs. Post Tx, etc.
Common in human trials to match on Age, Sex, Race,…    
… etc….
Sample 1, size n
Sample 2, size n
NOTE: Sample sizes are equal!
Since they are paired, subtract!
Treat as one sample of the normally distributed variable D = X – Y.

4. http://pages. stat. wisc

5. http://pages. stat. wisc