Economic Statistics ECN 521
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This 4 page Class Notes was uploaded by Ilene Heathcote on Wednesday October 21, 2015. The Class Notes belongs to ECN 521 at Syracuse University taught by Staff in Fall. Since its upload, it has received 37 views. For similar materials see /class/225648/ecn-521-syracuse-university in Economcs at Syracuse University.
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Date Created: 10/21/15
ECONOMIC STATISTICS Economics 521 The Central Limit Theorem If a random sample x1 x2 xn is observed quotom a population with mean u and variance 02 then for large n the sampling distribution of the sample mean X is approximately Nu ozn Hypothesis Testing We are going to reject a hypothesis if what we actually observe to occur would be very unlikely have low probability were the hypothesis true What we actually observe to occur must not be too nely detailed we must throw away some information We will pay attention to the values of sample statistics not to the full information in a sample Nor will we want to say that a single speci c observed value of a sample statistic is very unlikely because of course it would be We are going to reject a hypothesis if the value of a relevant sample statistic in an observed sample is in a region that is unlikely to occur were the hypothesis true For example suppose the hypothesis is that the mean of a population is 10 and that 0 is 2 If we now observe a sample mean to be 18 we will reject the hypothesis if it implies that means as large as or larger than 18 are very unlikely ie if Hi 218u 10and0 2isverysmall Null hypothesis H0 Alternative hypothesis H1 Decision Rule Type I error probability 06 Type 11 error probability 3 Example H0 ux 5 and 0x 5 H1 ux 7 and 0x 10 Decision rule Reject H0 and accept H1 just when X100 gt c STATO6wpd ECONOMIC STATISTICS Continuous random variables Probability density functions Probabilities as areas PX x The family of uniform density mctions On ab ux b a2 02X b a2 12 Standard normal density mction Standard normal table Freund and Simon Table 1 p 523 Standard Normal density Mean MK 0 Variance 02X 1 Standard deviation 0X 1 Transformed random variable A BX MW A Bux 02ABX B202 OABX l Blox The family of normal density mctions If X is normally distributed so is A BX If 2 is standard normally distributed then X p OZ has mean u and variance 02 If X is normally distributed and has mean u and variance 02 then X 7 100 is standard normal If X has a binomial distribution with parameters N and p and if both Np gt 5 and N1 p gt 5 then approximately X is distributed normally with mean Np and variance Np1 p SAMPLING DISTRIBUTION The probability distribution of a sample statistic from a random sample Random sample of size n from a population of size N All of the possible samples are equally likely Example The sample mean The Central Limit Theorem If a random sample x1 x2 xn is observed quotom a population with mean u and variance 02 then for large n the sampling distribution of the sample mean Xn is approximately Nu ozn
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