Applied Statistics and Probability for Engineers

11-7 PREDICTION OF NEW OBSERVATIONS 393

Let Y 0 be the future observation at xx 0 , and let given by Equation 11-32 be the es-

timator of Y 0. Note that the error in prediction

is a normally distributed random variable with mean zero and variance

because Y 0 is independent of If we use to estimate ^2 , we can show that

has a tdistribution with n 2 degrees of freedom. From this we can develop the following

prediction intervaldefinition.

Y 0  Yˆ 0

B

ˆ^2 c 1 

1

n

1 x 0    x 22

Sx x

d

Yˆ 0. ˆ^2

V 1 epˆ 2 V 1 Y 0   Yˆ 02 ^2 c 1 

1

n

1 x 0    x 22

Sx x

d

epˆY 0  Yˆ 0

Yˆ 0

A 100(1 ) % prediction intervalon a future observation at the value x 0 is

given by

(11-33)

The value yˆ 0 is computed from the regression model yˆ 0 ˆ 0 ˆ 1 x 0.

Y 0 yˆ 0 t 2, n     2

B

ˆ^2 c 1 

1

n

1 x 0    x 22

Sx x

d

yˆ 0     t 2, n    2

B

ˆ^2 c 1 

1

n

1 x 0    x 22

Sx x

d

Y 0

Definition

Notice that the prediction interval is of minimum width at and widens as

increases. By comparing Equation 11-33 with Equation 11-31, we observe that the prediction

interval at the point x 0 is always wider than the confidence interval at x 0. This results because

the prediction interval depends on both the error from the fitted model and the error associated

with future observations.

EXAMPLE 11-6 To illustrate the construction of a prediction interval, suppose we use the data in Example 11-1

and find a 95% prediction interval on the next observation of oxygen purity at x 0 1.00%.

Using Equation 11-33 and recalling from Example 11-5 that , we find that the

prediction interval is

Y 0 89.232.101

B

1.18 c 1 

1

20



1 1.00 1.1960 22

0.68088

d

89.23 2.101

B

1.18 c 1 

1

20



1 1.00 1.1960 22

0.68088

d

yˆ 0 89.23

x 0 x 0 x 0     x 0

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