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

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