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8-2 CONFIDENCE INTERVAL ON THE MEAN OF A NORMAL DISTRIBUTION, VARIANCE KNOWN 249

Confidence and tolerance intervals bound unknown elements of a distribution. In this
chapter you will learn to appreciate the value of these intervals. A prediction interval pro-
vides bounds on one (or more) future observations from the population. For example, a
prediction interval could be used to bound a single, new measurement of viscosity—another
useful interval. With a large sample size, the prediction interval for normally distributed data
tends to the tolerance interval in Equation 8-1, but for more modest sample sizes the predic-
tion and tolerance intervals are different.
Keep the purpose of the three types of interval estimates clear:
A confidence interval bounds population or distribution parameters (such as the mean
viscosity).
A tolerance interval bounds a selected proportion of a distribution.
A prediction interval bounds future observations from the population or distribution.

8-2 CONFIDENCE INTERVAL ON THE MEAN OF A NORMAL
DISTRIBUTION, VARIANCE KNOWN

The basic ideas of a confidence interval (CI) are most easily understood by initially consider-
ing a simple situation. Suppose that we have a normal population with unknown mean and
2
known variance . This is a somewhat unrealistic scenario because typically we know the
distribution mean before we know the variance. However, in subsequent sections we will
present confidence intervals for more general situations.

8-2.1 Development of the Confidence Interval and its Basic Properties

Suppose that X 1 , X 2 , p , X n is a random sample from a normal distribution with unknown
2
mean and known variance . From the results of Chapter 5 we know that the sample
2
mean X is normally distributed with mean and variance n . We may standardize X
by subtracting the mean and dividing by the standard deviation, which results in the
variable

X
Z (8-3)
1n

Now Z has a standard normal distribution.
A confidence interval estimate for is an interval of the form l u, where the end-
points l and u are computed from the sample data. Because different samples will produce
different values of l and u, these end-points are values of random variables L and U, respec-
tively. Suppose that we can determine values of L and U such that the following probability
statement is true:

P 5L U6 1 (8-4)

where 0 1. There is a probability of 1 of selecting a sample for which the CI will
x , X x , p ,
contain the true value of . Once we have selected the sample, so that X 1 1 2 2
X x , and computed l and u, the resulting confidence interval for is
n
n
l u (8-5)