7.4Estimating the Difference in Means of Two Normal Populations 257
are equal and letσ^2 denote their common value. Now, from Theorem 6.5.1 it follows
that
(n−1)
S 12
σ^2
∼χn^2 − 1
and
(m−1)
S 22
σ^2
∼χm^2 − 1
Also, because the samples are independent, it follows that these two chi-square ran-
dom variables are independent. Hence, from the additive property of chi-square random
variables, which states that the sum of independent chi-square random variables is also
chi-square with a degree of freedom equal to the sum of their degrees of freedom, it
follows that
(n−1)
S 12
σ^2
+(m−1)
S 22
σ^2
∼χn^2 +m− 2 (7.4.2)
Also, since
X−Y∼N
(
μ 1 −μ 2 ,
σ^2
n
+
σ^2
m
)
we see that
X−Y−(μ 1 −μ 2 )
√
σ^2
n
+
σ^2
m
∼N(0, 1) (7.4.3)
Now it follows from the fundamental result that in normal samplingXandS^2 are inde-
pendent (Theorem 6.5.1), thatX 1 ,S 12 ,X 2 ,S 22 are independent random variables. Hence,
using the definition of at-random variable (as the ratio of two independent random vari-
ables, the numerator being a standard normal and the denominator being the square root
of a chi-square random variable divided by its degree of freedom parameter), it follows
from Equations 7.4.2 and 7.4.3 that if we let
Sp^2 =
(n−1)S 12 +(m−1)S 22
n+m− 2
then
X−Y−(μ 1 −μ 2 )
√
σ^2 (1/n+1/m)
÷
√
Sp^2 /σ^2 =
X−Y−(μ 1 −μ 2 )
√
Sp^2 (1/n+1/m)
has at-distribution withn+m−2 degrees of freedom. Consequently,
P
{
−tα/2,n+m− 2 ≤
X−Y−(μ 1 −μ 2 )
Sp
√
1/n+1/m
≤tα/2,n+m− 2
}
= 1 −α