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1
T Test procedure compares the means of two groups or (one-sample) compares the means of a group with a constant.
Two Independent Samples
Notation
The following notation is used unless otherwise stated:
X (^) ki Value for^ i th case of group^ k w (^) ki Weight for^ i th case of group^ k n (^) k Number of cases in group^ k Wk Sum of weights of cases in group^ k
Basic Statistics
Means
X
X w
W k k
ki ki i
n
k
k
= = =
∑ (^1) 1 2,
Variances
S
X w X w W
k W
ki ki i
n ki ki i
n k
k
k k
2
2 1 1
2
F
H
GG
I
K
JJ
−
= =
∑ ∑
b g
Standard Errors of the Mean
SEM (^) k =S (^) k Wk
Differences of the Means for Groups 1 and 2
D = X 1 −X 2
Unpooled (Separate Variance) Standard Error of the Difference
S S
W
S
D W
2 1
22 2
The 95% confidence interval for mean difference is
D ± t (^) df ′SD
where t (^) df ′ is the upper 2.5% critical value for the t distribution with df (^) ′ degrees of freedom.
Pooled Standard Error of the Difference
S ′ = S +
D p W W
1 2
where the pooled estimate of the variance is
S W^ S^ W^ S
p W W 2 1 1
2 2 22
1 2
= −^ +^ −
b g b g
The Test for Equality of Variances
The Levene statistic is used and defined as
L
W W Z Z
w Z Z
k k k
ki ki k i
n
k
= k
=
= =
Â
ÂÂ
1
2
2 1 1
2
where
Z X X
Z
w Z
W
Z
W Z
W W
ki ki k
k
ki ki i
n
k
k k k
k
= -
=
=
=
=
Â
Â
1
1
2
1 2
The t Test for Paired Samples
Notation
The following notation is used unless otherwise stated:
X (^) i Value of variable^ X^ for case^ i Yi Value of variable^ Y^ for case^ i wi Weight for case^ i W Sum of the weights N Number of cases
Basic Statistics
Means
X w X W
Y w Y W
i i i
N
i i i
N
=
=
1
1
Variances
S
w X w X W X W
i i i
N i i i
N
2
2 1 1
2
F
H
GG
I
K
JJ
−
= =
∑ ∑
Similarly for SY^2.
Covariance between X and Y
S
W
XY X Y wk k k w^ k X^ k w Y^ W k
N k k k
N
k
N
−
F
H
GG
I
K
JJ
F
H
GG
I
K
JJ
F
H
GG
I
K
JJ = = =
∑ ∑ ∑
Difference of the Means
D = X −Y
Standard Error of the Difference
S (^) D = (^) eS (^) X^2 + S (^) Y^2 − 2 S (^) XYj W
The two-tailed significance level is based on
t r W r
with (^) bW − (^2) g degrees of freedom.
One-Sample t Test
Notation
The following notation is used unless otherwise stated:
N Number of cases Xi Value of variable X for case i ( i = 1, K, N ) w (^) i Weight for case i ( i = 1, K, N ). The weights must be positive. v Test value
Basic Statistics
Mean
X (^) W w Xi i i
N
=
∑
1
where W w (^) i i
N
=
∑ 1
is the sum of the weights.
Variance
S (^) X (^) W w (^) i X (^) i X i
N 2 2 1
= (^) − 1 ∑= d − i
Standard Deviation
S X = SX^2
Standard Error of the Mean
S X = S X / W
Mean Difference
D = X − v
The t value
t = D / S (^) X
with a W − 1 f degrees of freedom. A two-tailed significance level is printed.
100p% Confidence Interval for the Mean Difference a 0 < p < 1 f
CI = D ± t W − 1 , ( p + 1 ) / 2 SX
where t (^) W − 1 , ( p + 1 ) / 2 is the (^100) ca p + 1 f / (^2) h% percentile of a Student’s t distribution with a W − 1 f degrees of freedom.
References
Blalock, H. M. 1972. Social statistics. New York: McGraw-Hill.
