Introduction to Probability and Statistics for Engineers and Scientists

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