Page 295 - Applied statistics and probability for engineers

Section 8-1/Conidence Interval on the Mean of a Normal Distribution, Variance Known 273

Tolerance intervals are widely used and, as we will subsequently see, they are easy to calculate
for normal distributions.
Conidence 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 provides 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, but
for more modest sample sizes, the prediction and tolerance intervals are different.
Keep the purpose of the three types of interval estimates clear:
r A conidence interval bounds population or distribution parameters (such as the mean
viscosity).
A tolerance interval bounds a selected proportion of a distribution.
r A prediction interval bounds future observations from the population or distribution.
Our experience has been that it is easy to confuse the three types of intervals. For example, a
conidence interval is often reported when the problem situation calls for a prediction interval.

8-1 Confidence Interval on the Mean of a Normal

Distribution, Variance Known

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

8-1.1 DEVELOPMENT OF THE CONFIDENCE INTERVAL AND ITS BASIC PROPERTIES

1 ,
Suppose that X X 2 , ..., X n is a random sample from a normal distribution with unknown mean
2
μ and known variance σ . From the results of Chapter 5, we know that the sample mean X is
2
normally distributed with mean μ and variance σ / n. We may standardize X by subtracting
the mean and dividing by the standard deviation, which results in the variable
X − μ
Z = (8-1)
σ / n
The random variable Z has a standard normal distribution.
A conidence interval estimate for μ is an interval of the form l ≤ μ Ð 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, respectively. Suppose that we
can determine values of L and U such that the following probability statement is true:
{
P L ≤ μ ≤ U} = − α (8-2)
1
where 0 ≤ α ≤ 1. There is a probability of 1 − ≠ of selecting a sample for which the CI will con-
tain the true value of μ. Once we have selected the sample, so that X 1 = x X 2 = x 2 , ... , X n = x n ,
,
1
and computed l and u, the resulting conidence interval for μ is
l ≤ μ Ð u (8-3)

The end-points or bounds l and u are called the lower- and upper-conidence limits (bounds),
respectively, and 1 − ≠ is called the conidence coeficient.