Introduction to Probability and Statistics for Engineers and Scientists

9.4Statistical Inferences About the Regression Parameters 375

Prediction Interval for a Response at the Input Levelx 0

Based on the response valuesYicorresponding to the input valuesxi,i=1, 2,...,n:
With 100(1−a) percent confidence, the responseYat the input levelx 0 will be contained
in the interval

A+Bx 0 ±ta/2,n− 2

√[

n+ 1

n

+

(x 0 −x)^2

Sxx

]

SSR

n− 2

EXAMPLE 9.4f In Example 9.4c, suppose we want an interval that we can “be 95 percent
certain” will contain the height of a given male whose father is 68 inches tall. A simple
computation now yields the prediction interval

Y(68)∈67.568±1.050

or, with 95 percent confidence, the person’s height will be between 66.518 and
68.618. ■

REMARKS

(a)There is often some confusion about the difference between a confidence and a pre-
diction interval. A confidence interval is an interval that does contain, with a given degree
of confidence, a fixed parameter of interest. A prediction interval, on the other hand, is an
interval that will contain, again with a given degree of confidence, a random variable of
interest.
(b)One should not make predictions about responses at input levels that are far from
those used to obtain the estimated regression line. For instance, the data of Example 9.4c
should not be used to predict the height of a male whose father is 42 inches tall.

9.4.5 Summary of Distributional Results.................................

We now summarize the distributional results of this section.

Model: Y=α+βx+e, e∼N(0,σ^2 )

Data:(xi,Yi), i=1, 2,...,n

Inferences About Use the Distributional Result

β

√

(n−2)Sxx

SSr

(B−β)∼tn− 2

α

√√

√√

√

n∑(n−2)Sxx

i

x^2 iSSR

(A−α)∼tn− 2