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248 CHAPTER 8 STATISTICAL INTERVALS FOR A SINGLE SAMPLE

6. Explain the three types of interval estimates: confidence intervals, prediction intervals, and
tolerance intervals
7. Use the general method for constructing a confidence interval
CD MATERIAL
8. Use the bootstrap technique to construct a confidence interval

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8-1 INTRODUCTION

In the previous chapter we illustrated how a parameter can be estimated from sample data.
However, it is important to understand how good is the estimate obtained. For example, sup-
pose that we estimate the mean viscosity of a chemical product to be ˆ x 1000. Now
because of sampling variability, it is almost never the case that x . The point estimate says
ˆ
nothing about how close is to . 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 an example of an interval 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 confidence interval. We can-
not be certain that the interval contains the true, unknown population parameter—we only use
a sample from the full population to compute the point estimate and the interval. However,
the confidence interval is constructed so that we have high confidence that it does contain the
unknown population parameter. Confidence 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
1.96 , 1.96 (8-1)

However, this is not a useful tolerance interval because the parameters and are unknown.
x
Point estimates such as and s can be used in Equation 8-1 for and . 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 (8-2)

where k is an appropriate constant (that is larger than 1.96 to account for the estimation
error). As for a confidence interval, it is not certain that Equation 8-2 bounds 95% of the dis-
tribution, but the interval is constructed so that we have high confidence that it does.
Tolerance intervals are widely used and, as we will subsequently see, they are easy to cal-
culate for normal distributions.