Introduction to Probability and Statistics for Engineers and Scientists

9.4Statistical Inferences About the Regression Parameters 373

EXAMPLE 9.4e Using the data of Example 9.4c, determine a 95 percent confidence interval
for the average height of all males whose fathers are 68 inches tall.

SOLUTION Since the observed values are

n=10, x 0 =68, x=66.8, Sxx=171.6, SSR=1.49721

we see that
√
1
n

+

(x 0 −x)^2

Sxx

√

SSR

n− 2

=.1424276

Also, because

t.025,8=2.306, A+Bx 0 =67.56751

we obtain the following 95 percent confidence interval,

α+βx 0 ∈(67.239, 67.896) ■

9.4.4 Prediction Interval of a Future Response

It is often the case that it is more important to estimate the actual value of a future response
rather than its mean value. For instance, if an experiment is to be performed at temperature
levelx 0 , then we would probably be more interested in predictingY(x 0 ), the yield from this
experiment, than we would be in estimating the expected yield —E[Y(x 0 )]=α+βx 0.
(On the other hand, if a series of experiments were to be performed at input levelx 0 , then
we would probably want to estimateα+βx 0 , the mean yield.)
Suppose first that we are interested in a single value (as opposed to an interval) to use
as a predictor ofY(x 0 ), the response at levelx 0. Now, it is clear that the best predictor of
Y(x 0 ) is its mean valueα+βx 0. [Actually, this is not so immediately obvious since one
could argue that the best predictor of a random variable is (1) its mean — which minimizes
the expected square of the difference between the predictor and the actual value; or (2) its
median — which minimizes the expected absolute difference between the predictor and
the actual value; or (3) its mode — which is the most likely value to occur. However, as the
mean, median, and mode of a normal random variable are all equal — and the response is,
by assumption, normally distributed — there is no doubt in this situation.] Sinceαandβ
are not known, it seems reasonable to use their estimatorsAandBand thus useA+Bx 0
as the predictor of a new response at input levelx 0.
Let us now suppose that rather than being concerned with determining a single value
to predict a response, we are interested in finding a prediction interval that, with a given
degree of confidence, will contain the response. To obtain such an interval, letYdenote