ECON1203 Lecture Notes - Lecture 7: Point Estimation, Variance, Statistical Inference
6 - Estimation
Inferential statistics
• Extracting information about population parameters based on sample statistics
Past data idiates that 60% of passegers hoose eef oer hike
(a) What does that sample proportion tell us about the population proportion?
• In practical situations like this, parameters for the population are unknown
Using sample statistics to try to GUESS what the parameters are is usually the
only practical alternative
Process of inference in words
• Parameters describe key features of populations
• In practical situations, parameters are unknown
• Instead, a sample is drawn from the population to provide basic data
• These data are used to calculate various sample statistics
These sample statistics are used as estimators for population parameters
Estimation
• Statistic – any function of data in the sample
• Estimator – statistic whose purpose is to estimate a parameter or some function
thereof
• Point estimator – simply a formula (rule) for combining sample information to
produce a single number to estimate
• Estimators are random variables because they are functions of random variables, X1,
X2, …, Xn
• Examples of point estimators
Sample proportion is a point estimator for the population proportion
Sample mean is a point estimator for the population mean
Sample variance is a point estimator for the population variance
Properties of estimators
• Saple ea is a natural’ choice of estimator for the population mean
But there may be better estimators (i.e. s2 as estimator for 2 (divide by n – 1)
• Desirable properties of estimators
Unbiasedness – if we constructed it for each of many hypothetical samples of the
same size, will the estimator deliver the correct value (i.e. the value of the
parameter) on average
Consistency – as the sample size gets larger, does the probability that the
estiator deiates fro the paraeter y ore tha a sall aout eoe
smaller?
Relative efficiency – if there are two competing estimators of a parameter, does
the sampling distribution of one have less expected dispersion that that of the
other?
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Document Summary
Inferential statistics: extracting information about population parameters based on sample statistics. In practical situations like this, parameters for the population are unknown. Using sample statistics to try to guess what the parameters are is usually the only practical alternative. Process of inference in words: parameters describe key features of populations. In practical situations, parameters are unknown: these data are used to calculate various sample statistics. Instead, a sample is drawn from the population to provide basic data. These sample statistics are used as estimators for population parameters. X2, , xn: examples of point estimators. Sample proportion is a point estimator for the population proportion. Sample mean is a point estimator for the population mean. Sample variance is a point estimator for the population variance. Properties of estimators: sa(cid:373)ple (cid:373)ea(cid:374) is a (cid:858)natural" choice of estimator for the population mean. But there may be better estimators (i. e. s2 as estimator for 2 (divide by n 1: desirable properties of estimators.
