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Unformatted text preview: Sampling Theory Estimation Hypothesis Tests pvalues of tests Critical Values and Rejection Regions Introduction to Econometrics Chapter 3: Review of Statistics Geo rey Williams gwilliams@econ.rutgers.edu January 27, 2010 Geo rey Williams gwilliams@econ.rutgers.edu Introduction to Econometrics Chapter 3: Review of Statisti Sampling Theory Estimation Hypothesis Tests pvalues of tests Critical Values and Rejection Regions Random Sampling and the Sample Mean We typically do not get to observe the whole population of a random variable. Therefore we have to make do with a sample from the population. Suppose that we randomly choose N draws from the population. Let this sample be denoted f X 1 ;:::; X N g This is called a random sample from the population. Because the sampling is done on a random basis, X i , the i th draw from the population is a random variable. Also, because the sampling is random, any two draws from the population are independent of each other. Geo rey Williams gwilliams@econ.rutgers.edu Introduction to Econometrics Chapter 3: Review of Statisti Sampling Theory Estimation Hypothesis Tests pvalues of tests Critical Values and Rejection Regions Random Sampling and the Sample Mean As f X 1 ;:::; X N g are all drawn from the same population then the marginal distribution of each draw are identical. Therefore, we refer to this sample as an identically and independently distributed (i.i.d) sample . Geo rey Williams gwilliams@econ.rutgers.edu Introduction to Econometrics Chapter 3: Review of Statisti Sampling Theory Estimation Hypothesis Tests pvalues of tests Critical Values and Rejection Regions Sampling Distribution Because our sample is randomly chosen, then each sample is a random variable. This means that any function of the sample will also be a random variable. The most obvious function of the sample that we are interested in is the sample average . The sample average is de ned as X = 1 N N X i = 1 X i : As is the case for any random variable we are interested in knowing the mean, variance and distribution of the sample average Geo rey Williams gwilliams@econ.rutgers.edu Introduction to Econometrics Chapter 3: Review of Statisti Sampling Theory Estimation Hypothesis Tests pvalues of tests Critical Values and Rejection Regions Properties of the Sample Average Let f X 1 ;:::; X N g be an identically and independently distributed sample from a distribution with mean and variance 2 . Then 1 E ( X ) = 2 var ( X ) = 2 N 3 std dev ( X ) = p N 4 if X i N ( ; 2 ) for each i then X N ; 2 N Geo rey Williams gwilliams@econ.rutgers.edu Introduction to Econometrics Chapter 3: Review of Statisti Sampling Theory Estimation Hypothesis Tests pvalues of tests Critical Values and Rejection Regions Large Sample Properties The last result above showed that if the sample comes from a Normal distribution then the sample mean has a Normal distribution. This is the case for any size sample and is referred to as the nite sample or exact distribution of the...
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This note was uploaded on 02/29/2012 for the course 320 322 taught by Professor Macrowilliams,microyoshi during the Fall '10 term at Rutgers.
 Fall '10
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