Page 294 - Applied statistics and probability for engineers

272 Chapter 8/Statistical intervals for a single sample

impact energy, and only 5% of the time would the interval be in error. In this chapter, you will
learn how to construct conidence intervals and other useful types of statistical intervals for
many important types of problem situations.

Learning Objectives
After careful study of this chapter, you should be able to do the following:

1. Construct confidence intervals on the mean of a normal distribution, using either the normal
distribution or the t distribution method
2. Construct confidence intervals on the variance and standard deviation of a normal distribution
3. Construct confidence intervals on a population proportion
4. Use a general method for constructing an approximate confidence interval on a parameter
5. Construct prediction intervals for a future observation
6. Construct a tolerance interval for a normal population
7. Explain the three types of interval estimates: confidence intervals, prediction intervals, and tolerance
intervals

In the previous chapter, we illustrated how a point estimate of a parameter can be estimated
from sample data. However, it is important to understand how good the estimate obtained
is. For example, suppose that we estimate the mean viscosity of a chemical product to be
.
ˆ μ = x = 1000 Now because of sampling variability, it is almost never the case that the true
mean μ is exactly equal to the estimate x. The point estimate says nothing about how close ˆ μ
is to μ. Is the process mean likely to be between 900 and 1100? Or is it likely to be between
990 and 1010? The answer to these questions affects our decisions regarding this process.
Bounds that represent an interval of plausible values for a parameter are examples of an inter-
val estimate. Surprisingly, it is easy to determine such intervals in many cases, and the same
data that provided the point estimate are typically used.
An interval estimate for a population parameter is called a conidence interval. Informa-
tion about the precision of estimation is conveyed by the length of the interval. A short interval
implies precise estimation. We cannot be certain that the interval contains the true, unknown
population parameter—we use only a sample from the full population to compute the point
estimate and the interval. However, the conidence interval is constructed so that we have high
conidence that it does contain the unknown population parameter. Conidence intervals are
widely used in engineering and the sciences.
A tolerance interval is another important type of interval estimate. For example, the
chemical product viscosity data might be assumed to be normally distributed. We might like
to calculate limits that bound 95% of the viscosity values. For a normal distribution, we know
that 95% of the distribution is in the interval
μ −1.96σ μ −19.6σ
,
However, this is not a useful tolerance interval because the parameters μ and σ are unknown.
Point estimates such as x and s can be used in the preceding equation for μ and σ. However,
we need to account for the potential error in each point estimate to form a tolerance interval
for the distribution. The result is an interval of the form

x − ks x + ks
,
where k is an appropriate constant (that is larger than 1.96 to account for the estimation error).
As in the case of a conidence interval, it is not certain that the tolerance interval bounds 95%
of the distribution, but the interval is constructed so that we have high conidence that it does.