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Lecture 8

# STAT 2507 Lecture Notes - Lecture 8: Bias Of An Estimator, Interval Estimation, Point Estimation

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Chapter 8

Large Sample Estimation

In the previous chapters we have learned everything we need to understand statistical inference

and how it can be applied in practical situations. In this chapter we will learn more about

estimation and how to infer about population parameters using point estimators. Statistical

inference Statistical inference is the process employed to make decisions or predictions about

parameters. Examples of population parameters: Population mean, µ Population standard

deviation, σ Population proportion, p

Examples of applications of inference:

The prediction of short- and long-term interest rates

The forecasting of the behaviour of the stock market

A consumer wants to estimate the selling price of her house before putting it on the market

Deciding on the effectiveness of a new drug

Statistical Inference Definitions:

Estimation: is the process of estimating or predicting the value of a parameter of interest.

Hypothesis Testing: is the process of making decision about the value of a parameter based on

some preconceived idea about what its value might be.

Types of Estimators

Definition: an estimator is a rule, usually expressed as a formula, that tells us how to calculate an

estimate based on the information in the sample.

Estimators are used in two ways.

Point estimation: Based on sample data, a single number is calculated to estimate a population

parameter. The rule or formula that describes this calculation is called the point estimator, and

the resulting number is called a point estimate

Interval estimation: Based on sample data, two numbers are calculated to form an interval

within which the parameter is expected to lie. The rule or formula used in the calculation is

called an interval estimator, and the pair of numbers obtained is called an interval estimate or

confidence interval.

Point Estimation

In practice, there can be more than one point estimator that can be used to estimate a

population parameter of interest. The choice of which point estimator would depend on the

behaviour of the point estimator and its sampling distribution.

The first characteristic that is desired to have in a point estimator is that its sampling

distribution centres around the true value of the parameter. An estimator that has this

characteristic is called unbiased.

Point Estimation-Unbiased Estimators An estimator is said to be unbiased if its expected value

is equal to the population parameter it estimates. That is, the long-run average value of the

estimator is equal to the population parameter. NOTE: Every estimator we have seen so far is

unbiased:

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