Point and Interval Estimates of the Population Mean

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.

The calculated value of the sample mean in a given sample, used as an estimate of the populationmean, is called a point estimate of the population mean.

Three desirable properties of estimators:

Unbiasedness.

An unbiased estimator is one whose expected value equals the parameter it is intended to estimate.

Efficiency.

An unbiased estimator is efficient if no other unbiased estimator of the same parameter has a sampling distribution with smaller variance.

Consistency.

A consistent estimator is one for which the probability of estimates close to the value of the population parameter increases as sample size increases.

As sample size n goes to infinity, the standard error,  , goes to 0 and its sampling distribution becomes concentrated right over the value of population mean, μ.

[Practice Problems] If an estimator is consistent, an increase in sample size will increase the:

A.       accuracy of estimates.

B.        efficiency of the estimator.

C.        unbiasedness of the estimator.

[Solutions] A

A consistent estimator is one for which the probability of estimates close to the value of the population parameter increases as sample size increases. More specifically, a consistent estimator’s sampling distribution becomes concentrated on the value of the parameter it is intended to estimate as the sample size approaches infinity.