Samplingdistnotes_Fall2009

Samplingdistnotes_Fall2009 - SAMPLING DISTRIBUTION...

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SAMPLING DISTRIBUTION (REFERENCE: TEXTBOOK CHAPTER 3) Population: a collection of all units of interest in a study (e.g. all students in a ECON2P91) Sample: part of the population (e.g. 20 students from ECON2P91 class-assumed to be random) Suppose that we want to determine the average income of all students in ECON2P91 last year. We could get the exact value (i.e. population mean) by collecting information on all incomes. However, doing so can be expensive due to, say, cost and time constraints; may be impossible if the population is infinite; or may be impractical if the study involves destructive testing. A more convenient alternative is to take a random sample of, say, 20 students from the class and use information about their average income (i.e. sample mean) to deduce information about the average income for the entire class (i.e. population mean). The problem is that the value of the sample mean we get depends on the sample we end up selecting. Therefore, the sample mean has a probability distribution . This probability distribution is called the sampling distribution of the sample mean . By definition a sampling distribution is a probability distribution of a sample statistic ( a statistic is any numerical quantity we obtain using sample observations; in contrast a parameter is a numerical quantity that we obtain using all the population observations i.e. the population mean is a parameter and the sample mean is a statistic. A sample statistic is therefore used to deduce information about the population parameter. Interesting properties of the sampling distribution of the sample mean Suppose that we take a random sample of n observations from a given population. If is the sample mean (i.e. ) ; is the (unknown) population mean; is the population variance, and hence, is the population standard deviation (the standard deviation is the square root of the variance). The following results are well known in the statistics literature: 1. (i.e. the mean (expected value) of sampling distribution of the mean is always equal to the population. 2. (i.e. the variance of the sampling distribution of the mean is equal to population variance divided by the sample size (n). 3. (i.e. the standard deviation of the sampling distribution of the mean (i.e. the square root of the variance of the sampling distribution of the mean) is equal to the population standard deviation divided by the square root of the sample size. The standard deviation for the mean is also called the standard error of the mean i.e. the standard error of the mean denoted . 4. If the population from which the sample is taken is normally distributed the sampling distribution of the mean is also normally distributed. 5. If the population from which the sample is taken is not normally distributed the sampling distribution of the mean is
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Samplingdistnotes_Fall2009 - SAMPLING DISTRIBUTION...

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