Stat 333 Spring 2004 1/30/2004
1 Confidence Interval for β1
The variance and standard deviation for b1are given by the following:
V(b1) = σ2
SXX
sd(b1) = σ
√SXX
est.sd(b1) = s
√SXX
Note that b1is the estimator of β1such that b1=SXY
SXX
. When constructing the confidence interval
for β1, we should know the following facts:
(1) β1is the parameter of interest and it is fixed, not random.
(2) b1is a random variable and it has the normal distribution with mean β1and variance σ2
SXX
.
(3) From (2), we know that b1−β1
σ/√SX X
has the standard normal distribution, N(0,1). Since σ2
is unknown, we use s2to estimate σ2, where s2= MSE = SSE/(n−2). Therefore, b1−β1
s/√SXX
has the t-distribution with (n-2) degrees of freedom, i.e.,
b1−β1
s/√SXX ∼tn−2
With these facts, we have
P −t(n−2,1−α
2)<b1−β1
s/√SXX
< t(n−2,1−α
2)!= 1 −α
That is,
P b1−t(n−2,1−α
2)·s
√SXX
< β1< b1+t(n−2,1−α
2)·s
√SXX != 1 −α
So the 100(1 −α)% confidence interval for β1is
b1±t(n−2,1−α
2)·s
√SXX