POINT AND INTERVAL ESTIMATES OF THE POPULATION MEAN

POINT AND INTERVAL ESTIMATES OF THE POPULATION MEAN

Statistical inference traditionally consists of two branches, hypothesis testing and estimation. Hypothesis testing addresses the question “Is the value of this parameter (say, a population mean) equal to some specific value (0, for example)?” In this process, we have a hypothesis concerning the value of a parameter, and we seek to determine
whether the evidence from a sample supports or does not support that hypothesis.
We discuss hypothesis testing in detail in the reading on hypothesis testing. The second branch of statistical inference, and the focus of this reading, is estimation.
Estimation seeks an answer to the question “What is this parameter’s (for example, the population mean’s) value?” In estimating, unlike in hypothesis testing, we do not start with a hypothesis about a parameter’s value and seek to test it. Rather, we try to make the best use of the information in a sample to form one of several types
of estimates of the parameter’s value. With estimation, we are interested in arriving at a rule for best calculating a single number to estimate the unknown population parameter (a point estimate). Together with calculating a point estimate, we may also be interested in calculating a range of values that brackets the unknown population
parameter with some specified level of probability (a confidence interval). In Section 4.1 we discuss point estimates of parameters and then, in Section 4.2, the formulation of confidence intervals for the population mean.
Point Estimators An important concept introduced in this reading is that sample statistics viewed as
formulas involving random outcomes are random variables. The formulas that we use to compute the sample mean and all the other sample statistics are examples of estimation formulas or estimators. The particular value that we calculate from sample observations using an estimator is called an estimate. An estimator has a sampling
distribution; an estimate is a fixed number pertaining to a given sample and thus has no sampling distribution. To take the example of the mean, the calculated value of the sample mean in a given sample, used as an estimate of the population mean, is called a point estimate of the population mean. As Example 3 illustrated, the formula for
the sample mean can and will yield different results in repeated samples as different samples are drawn from the population.

In many applications, we have a choice among a number of possible estimators for estimating a given parameter. How do we make our choice? We often select estimators because they have one or more desirable statistical properties. Following is a brief description of three desirable properties of estimators: unbiasedness (lack of
bias), efficiency, and consistency

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