Paired-Samples t-test
10/13
Independence
• Knowing one variable tells nothing about
another
– Probabilities: p(X) unchanged by value of Y
– Measures from unrelated people or events
• Critical question for inferential statistics
– Affects sampling distributions
– Example: Household income
• Sample 100 unrelated people
• Sample 50 couples (100 people total)
Independent vs. Paired Samples
• Independent-sample t-test assumes no relation
between Sample A and Sample B
– Unrelated subjects, randomly assigned
– Necessary for standard error to be correct
• Sometimes samples are paired
– Each score in Sample A goes with a score in Sample B
– Before vs. after, husband vs. wife, matched controls
– Paired-samples t-test
Paired-samples t-test
• Data are pairs of scores, (XA, XB)
– Form two samples, XA and XB
– Samples are not independent
• Same null hypothesis: mA = mB
– Equivalent to mean(XA – XB) = 0
• Approach
– Compute difference scores, X = XA – XB
– One-sample t-test on difference scores, with m0 = 0
Example
• Breath holding underwater vs. on land
– 8 subjects
– Water:
XA = [54, 98, 67, 143, 82, 91, 129, 112]
– Land:
XB = [52, 94, 69, 139, 79, 86, 130, 110]
• Difference: X = [2, 4, -2, 4, 3, 5, -1, 2]
–
–
–
–
–
Mean: M = SX/n = 17/8 = 2.13
Sample variance (MSE): s2 = S(X-M)2/(n-1) = 6.13
Standard error: sM = s/√n = √6.13/√8 = .88
t = M/sM =2.13/.88 = 2.43
p(|t7| ≥ 2.43) = .046
• Reliably longer underwater
Comparison of t-tests
Samples
One
Data
t
X
M  m0
sM
Standard Error
s

n
MSE
df
 X  M 
2
MSE
n
s2 
df
n-1
 X A  M A 
2
Indep.
Paired

XA, XB

X = XA - XB
M
MA 
B
sM
A
MSE
M B

M
sM

1
nA
1
 nB

 X B  M B 
2
nA + nB – 2
df
s

n
MSE

n
 X  M 
2
s 
2
df
n-1