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Section 8-7/Tolerance and Prediction Intervals 297

estimated by using bootstrap samples. The procedure would consist of taking n B bootstrap
samples, calculating the bootstrap estimates θ θ 2 … , θ n B and θ , and then computing the
ˆ B
ˆ B
ˆ B
B
,
,
1
differences θ i − θ B , i = 1 , ,… ,n . The α/2 smallest and largest of these differences are the
ˆ B
2
B
estimates of the percentiles required to construct the bootstrap CI.
8-7 Tolerance and Prediction Intervals
8-7.1 PREDICTION INTERVAL FOR A FUTURE OBSERVATION
In some problem situations, we may be interested in predicting a future observation of a vari-
able. This is a different problem than estimating the mean of that variable, so a conidence
interval is not appropriate. In this section, we show how to obtain a 100(1 – α)% prediction
interval on a future value of a normal random variable.
Suppose that X , X , …, X is a random sample from a normal population. We wish to pre-
1 2 n
dict the value X , a single future observation. A point prediction of X is X, the sample
n+1
n+1
mean. The prediction error is X n+ − X. The expected value of the prediction error is
1
(
E X n + − ) = μ − μ = 0
X
1
and the variance of the prediction error is
(
V X n + − ) = σ 2 + σ 2 = σ 2 ⎛ ⎜ 1 + 1 ⎞ ⎟
X
1
n ⎝ n⎠
because the future observation X is independent of the mean of the current sample X. The
n+1
prediction error X – X is normally distributed. Therefore,
n+1
− X
Z = X +n 1
σ 1 + 1
n
has a standard normal distribution. Replacing σ with S results in
− X
T = X +n 1
S 1 + 1
n
which has a t distribution with n – 1 degrees of freedom. Manipulating T as we have done pre-
viously in the development of a CI leads to a prediction interval on the future observation X .
n+1
Prediction Interval
A 100(1 – α)% prediction interval (PI) on a single future observation from a normal
distribution is given by

1 1
x − t / ,nα 2 −1 s 1 + ≤ X n ≤ x + t / ,nα 2 −1 s 1 + (8-28)
+1
n n

The prediction interval for X will always be longer than the conidence interval for μ
n+1
because more variability is associated with the prediction error than with the error of estimation.
This is easy to see because the prediction error is the difference between two random variables
(X n+ − X ), and the estimation error in the CI is the difference between one random variable and
1
a constant (X − μ ). As n gets larger (n → ∞ ), the length of the CI decreases to zero, essentially
becoming the single value μ, but the length of the PI approaches 2z α σ. So as n increases, the
2 /
uncertainty in estimating μ goes to zero, although there will always be uncertainty about the
future value X , even when estimating any of the distribution parameters is not necessary.
n+1