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Chapter 6

Interval estimation

6.1. Introduction

The problem of statistical inference arises when we wish to make generalization about a population when only a sample will be available. Once a sample is observed, its main features can be determined by the methods of descriptive summary discussed in previous chapters. Our principal concern is with not just the particular data set, but what can be said about the population based on the information extracted from analyzing the sample data.

Statistical inference deals with drawing conclusions about population parameters from an analysis of the sample data.

The value(s) assigned to a population parameters based on the value of a sample statistic is called an estimate of the population parameters.

For example, suppose the manager selects a sample of 50 new employees and finds that the mean time taken to learn the job for these employees is 10 hours. If manager assigns this value to the population mean, then 10 hours will be called an estimate of . Thus, the sample mean is an estimator of the population mean , and the sample proportion is an estimator of the population proportion p.

An estimate may be a point estimate or an interval estimate.

Definition:

The value of a sample statistic that is used to estimate population parameters is called a point estimate.

Each sample taken from a population is expected to yield a different value of the sample statistics. Thus, the value assigned to a population parameter based on the point estimate depends on which of the sample is drawn. Consequently, the point estimate assigns a value to a population parameters almost always differs from the true value of the population parameters.

In the case of interval estimation, instead of assigning a simple value to a population parameter, an interval is constructed around the point estimate and then a probability statement that this interval contains the corresponding population parameter is made.

Definition:

In interval estimation, an interval is constructed around the point estimate, and it is stated that this interval likely to contain the corresponding population parameter.

6.2. Confidence interval and confidence level

Since interval estimators have been described as “likely” to contain the true, but unknown value of the population parameters, it is necessary to phrase such term as probability statement.

Suppose that a random sample is selected and based on the sample information, it is possible to find two random variables a and b. Then interval extending from a and b either includes the population parameter or it does not contain population parameter. However, suppose that the random samples are repeatedly selected from the population and similar intervals are found. In the long run, a certain percentage of this interval will contain the unknown value. According to the frequency concept of probability, an interpretation of such intervals follows:

If the population is repeatedly samples and intervals calculated, then in the long run 90% (or some other percentages) of the intervals would contain the true value of the unknown parameter. The interval from a and b is said to be 90% (or some other percentages) confidence interval estimator for population parameters.

Definition:

Let be unknown parameter. Suppose that based on sample information, random variables a and b are found such that

,

where -is any number between 0 and 1.

The interval from a to b is called confidence interval for .

The quantity is called the confidence level of the interval.

If the population were repeatedly sampled a very large number of times, the true value of the parameter would be contained in of intervals calculated this way.

The confidence interval calculated in this way is written as with confidence.

Let us find the confidence intervals with any required confidence level , where is any number such that .

We will use the notation for the number such that

.

A notation indicates the value in the standard normal table cuts off a right tail area of . (Fig. 6.1).

For example, if , then .

So,

and from the standard normal distribution table we obtain .

Therefore

.

Now suppose that a confidence interval is required. (Fig.6.2).

We have

By the symmetry about the mean

.

And it follows that

,

where the random variable Z follows a standard normal distribution.

Example:

Find the value of if .

Solution:

.

.

and

(Fig. 6.3).

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