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7-1 INTRODUCTION 221

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7-1 INTRODUCTION

The field of statistical inference consists of those methods used to make decisions or to draw
conclusions about a population. These methods utilize the information contained in a sample
from the population in drawing conclusions. This chapter begins our study of the statistical
methods used for inference and decision making.
Statistical inference may be divided into two major areas: parameter estimation and
hypothesis testing. As an example of a parameter estimation problem, suppose that a structural
engineer is analyzing the tensile strength of a component used in an automobile chassis. Since
variability in tensile strength is naturally present between the individual components because of
differences in raw material batches, manufacturing processes, and measurement procedures (for
example), the engineer is interested in estimating the mean tensile strength of the components.
In practice, the engineer will use sample data to compute a number that is in some sense a rea-
sonable value (or guess) of the true mean. This number is called a point estimate. We will see
that it is possible to establish the precision of the estimate.
Now consider a situation in which two different reaction temperatures can be used in a
chemical process, say and . The engineer conjectures that results in higher yields than
t 2
t 1
t 1
does t 2 . Statistical hypothesis testing is a framework for solving problems of this type. In this
case, the hypothesis would be that the mean yield using temperature is greater than the mean
t 1
yield using temperature t 2 . Notice that there is no emphasis on estimating yields; instead, the
focus is on drawing conclusions about a stated hypothesis.
Suppose that we want to obtain a point estimate of a population parameter. We know that
before the data is collected, the observations are considered to be random variables, say
X , X , p , X . Therefore, any function of the observation, or any statistic, is also a random
1
2
n
variable. For example, the sample mean X and the sample variance S 2 are statistics and they
are also random variables.
Since a statistic is a random variable, it has a probability distribution. We call the proba-
bility distribution of a statistic a sampling distribution. The notion of a sampling distribution
is very important and will be discussed and illustrated later in the chapter.
When discussing inference problems, it is convenient to have a general symbol to represent

the parameter of interest. We will use the Greek symbol (theta) to represent the parameter. The
objective of point estimation is to select a single number, based on sample data, that is the most

plausible value for . A numerical value of a sample statistic will be used as the point estimate.
In general, if X is a random variable with probability distribution f 1x2 , characterized by
is a random sample of size n from X, the
1
the unknown parameter , and if X , X 2 , p , X n
ˆ
statistic h1X , X , p , X 2 is called a point estimator of . Note that ˆ is a random vari-
n
1
2
ˆ
able because it is a function of random variables. After the sample has been selected, takes
ˆ

on a particular numerical value called the point estimate of .
Definition
A point estimate of some population parameter is a single numerical value ˆ of a
ˆ
ˆ

statistic . The statistic is called the point estimator.