## Introduction

In this chapter, we begin to explore the possibility of making inferential statements about a population, based on the information contained in a random sample. We will focus attention on specific characteristics, or parameters, of the population. Parameters of interest might include the population mean or variance, or the proportion of population members possessing some specific attribute. For example, we may want to make inferences about

1. The mean income of all families in a particular neighborhood

2. The variation in the impurity levels in batches of a manufactured chemical

3. The proportion of a corporation's employees favoring the introduction of a modified bonus scheme

Any inference drawn about the population will, of necessity, be based on sample statistics—that is, on functions of the sample information. The choice of appropriate statistics will depend on which population parameter is of interest. The true parameter will be unknown, and one objective of sampling could be to estimate its value.

### Definitions

An estimator of a population parameter is a random variable that depends on the sample information and whose realizations provide approximations to this unknown parameter. A specific realization of that random variable is called an estimate.

To clarify the distinction between the terms estimator and estimate, consider the estimation of the mean income of all families in a neighborhood, based on a random sample of twenty families. It seems reasonable to base our conclusions on the sample mean income, so we say that the estimator of the population mean is the sample mean. Suppose that, having obtained the sample, we find that the average income of the families in the sample is \$49,356. Then the estimate of the population mean family income is \$49,356. Notationally, we have already made this distinction, using X to denote the random variable and x a specific realization.

In discussing the estimation of an unknown population parameter, two possibilities must be considered. First, we could compute from the sample a single number as "representative," or perhaps "most representative." The estimate \$49,356 for the neighborhood mean family income is an example of such an estimate. Alternatively, we could try to find an interval, or range, that we are fairly sure contains the true parameter. In this chapter, we consider the first type of estimation problem, postponing until Chapter 8 a discussion of interval estimation.

### Definitions

A point estimator of a population parameter is a function of the sample information that yields a single number. The corresponding realization is called the point estimate of the parameter.

In the neighborhood family income example, the parameter to be estimated is the population mean family income. The point estimator used is the sample mean, and the resulting point estimate is \$49,356.

For purposes of illustration, we will discuss in this chapter four point estimators, all of which were met in Chapter 6. These are the sample mean, variance, standard deviation, and proportion. Table 7.1 summarizes the notation we have used.

TABLE 7.1 Notation for population parameters, point estimators, and estimates

POPULATION PARAMETER ESTIMATOR ESTIMATE

Standard deviation (ct>) sx

Proportion (p) px P*

EXAMPLE Price-earnings ratios for a random sample of ten stocks traded on the New York

7 j Stock Exchange on a particular day were

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