Confidence Interval Estimation: One Population

Introduction:

What is the average number of gallons of orange juice sold weekly by a local grocery store?

Management of this grocery store could use an estimate of the average weekly demand for

orange juice (milk, bread, or fresh fruit) to improve the ordering process, reduce waste

(such as spoiled fruit), reduce costs, and in-crease profits.

We present two estimation procedures in this chapter. First, we estimate an unknown

population parameter by a single number called a point estimate. Properties of this point

estimate are considered in Section 7.1. For most practical problems, a point estimate alone is

not adequate. A more complete understanding of the process that generated the population

also requires a measure of variability. Next we discuss a procedure that takes into account this

variation by establishing an interval of values, known as a confidence interval, which is likely

to include the quantity.

Properties of point estimators:

•  Estimator and estimate: 

An estimator of a population parameter is a random variable that depends on the sample information; its value provides approximations of this unknown parameter. A specific value of that random variable is called an estimate.

•  Point Estimator and Point Estimate: 

Consider a population parameter such as the population mean m or the population proportion P. A point estimator of a population parameter is a function of the sample information that produces a single number called a point estimate. 

For example, the sample mean X is a point estimator of the population mean, m, and the value that X assumes for a given set of data is called the point estimate, x.

• Unbiased Estimator: 

In searching for an estimator of a population parameter, the first property an estimator should possess is unbiasedness.

• Bias: 

Unbiasedness alone is not the only desirable characteristic of an estimator. There may be several unbiased estimators for a population parameter. For example, if the population is normally distributed, both the sample mean and the median are unbiased estimators of the population mean.

Most efficient: 

In many practical problems, different unbiased estimators can be obtained, and some method of choosing among them needs to be found. In this situation it is natural to prefer the estimator whose distribution is most closely concentrated about the population parameter being estimated. Values of such an estimator are less likely to differ, by any fixed amount, from the parameter being estimated than are those of its competitors. Using variance as a measure of concentration, the efficiency of an estimator as a criterion for prefer-ring one estimator to another estimator is introduced.