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8-6 A PREDICTION INTERVAL FOR A FUTURE OBSERVATION 269

has a standard normal distribution. Replacing with S results in

X
X n 1
T
1
1
S
B n
which has a t distribution with n 1 degrees of freedom. Manipulating T as we have done previ-
ously in the development of a CI leads to a prediction interval on the future observation X n 1 .

Definition
A 100(1 )% prediction interval on a single future observation from a normal
distribution is given by

1 1
x t 2,n 1 s B 1 n X n 1 x t 2,n 1 s B 1 n (8-29)

The prediction interval for X n 1 will always be longer than the confidence interval for
because there is more variability associated with the prediction error than with the error of es-
timation. This is easy to see because the prediction error is the difference between two random
variables (X n 1 X ), and the estimation error in the CI is the difference between one random
variable and a constant (X ). As n gets larger (n S
), the length of the CI decreases to
zero, essentially becoming the single value , but the length of the prediction interval
approaches 2z 2 . So as n increases, the uncertainty in estimating goes to zero, although
there will always be uncertainty about the future value X n 1 even when there is no need to
estimate any of the distribution parameters.
EXAMPLE 8-8 Reconsider the tensile adhesion tests on specimens of U-700 alloy described in Example 8-4.
x
The load at failure for n 22 specimens was observed, and we found that 13.71 and
s 3.55. The 95% confidence interval on was 12.14 15.28. We plan to test
a twenty-third specimen. A 95% prediction interval on the load at failure for this specimen is

1 1
x t 2, n 1 s B 1 n X n 1 x t 2,n 1 s B 1 n

1 1
1 1
B 22 B 22
23
13.71 12.08023.55 X 13.71 12.08023.55
6.16 X 21.26
23
Notice that the prediction interval is considerably longer than the CI.

EXERCISES FOR SECTION 8-6

8-49. Consider the tire-testing data described in Exercise 8-22. 8-50. Consider the Izod impact test described in Exercise 8-23.
Compute a 95% prediction interval on the life of the next tire of Compute a 99% prediction interval on the impact strength of
this type tested under conditions that are similar to those em- the next specimen of PVC pipe tested. Compare the length of
ployed in the original test. Compare the length of the prediction the prediction interval with the length of the 99% CI on the
interval with the length of the 95% CI on the population mean. population mean.