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lecturenotes08 - ECON 41: Statistics for Economists Lecture...

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ECON 41: Statistics for Economists Lecture Notes 0 8 Chunming Yuan econ41yuan@gmail.com Economics Department, UCLA
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In previous lectures we considered estimation and testing hypotheses about the population mean and proportion of a single population. In this lecture we will consider estimation and testing hypotheses i.e. making inferences about the di f erence between two population means and between two population proportions . For example we may want to estimate the di f erence in the mean wage between men and women and then construct a con f dence interval for it. Or we may want to test the hypothesis that the percentage of votes the demo- cratic candidate received in California is di f erent that the democratic candidate in New York.
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Independent vs Dependent Samples Two samples drawn from two populations are independent if the selection of one sample from one population does not a f ect the selection of the second sample from the second population. Otherwise the samples are dependent . Suppose we want two estimate the di f erence between the mean salaries of all male and all female executives. The samples of male executives and females executives that we draw to estimate this di f erence are independent. Supposewewan ttoest imatethed i f erence between the mean weights of all
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Notation μ 1 = Mean of population 1 μ 2 = Mean of population 2 σ 1 = Standard deviation of population 1 σ 2 = Standard deviation of population 2 n 1 = Sample size of population 1 n 2 = Sample size of population 2 x 1 = Sample mean of population 1 x 2 = Sample mean of population 2
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Inference about μ 1 μ 2 for Independent Samples Case 1: Normal populations, known σ 1 and σ 2 Inference on the di f erence between the population means from two normal populations is based on the fact that X 1 X 2 N Ã μ 1 μ 2 , σ 2 1 n 1 + σ 2 2 n 2 ! Let σ X 1 X 2 denote the standard deviation of the di f erence of the two sample averages: σ X 1 X 2 = v u u t σ 2 1 n 1 + σ 2 2 n 2
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Note: This formula obtains for the standard deviation of the di f erenced av- erages only if the two samples are independent . The standardized di f erence in the two sample means will therefore be standard normally distributed: ³ X 1 X 2 ´ ( μ 1 μ 2 ) r σ 2 1 n 1 + σ 2 2 n 2 N (0 , 1) Even if the populations are not normal, if the sample sizes are large we use the fact that the standardized di f erence in the two sample averages is approx- imately distributed according to the standard normal distribution.
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(1 a )100% Con f dence Interval for μ 1 μ 2 Given the con f dence level (1 α ) ,wecan f nd z α such that P z α < ³ X 1 X 2 ´ ( μ 1 μ 2 ) σ X 1 X 2 <z α =1 α Rearranging we obtain
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This note was uploaded on 09/03/2009 for the course ECON 41 taught by Professor Guggenberger during the Summer '07 term at UCLA.

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lecturenotes08 - ECON 41: Statistics for Economists Lecture...

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