L8 Comparing Population Means and Variances

L8 Comparing Population Means and Variances - Comparing...

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Comparing Population Means and Variances. Testing for the difference between means in two populations. Given two independent random samples X 1i , i = 1,. .,N 1 and X 2j , j=1,. .,N 2 (the independence is both between and within the samples) from two populations which are respectively N(! 1 ," 1 2 ) and N(! 2 ," 2 2 ) the sample means, which will be independent of each other, will be distributed: 22 12 11 2 2 , ,; XN NN ±± ²² ³´ ³ µ ¶· ¸¹ ¸ ´ · ¹ (Note that if the distributions are not normal the sample means will still be approximately normal by the Central Limit Theorem). Since it is a property of the normal distribution that sums and differences of normal random variables are also normal with a variance equal to the sum of the variances in both cases (we shall demonstrate this in a later chapter) it follows that: 1 2 , XX N µº º » ± which, when the variances are known, can be used to test hypotheses about ! 1 - ! 2 in a fashion similar to the tests for a single population mean which were outlined in course notes 5. This follows since transforming to a Z variable we have: ¼½ 1 2 (0,1) ZN µº º º ¾ » ± which for hypothesized values of !
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This note was uploaded on 03/23/2011 for the course ECONOMICS 209 taught by Professor Kambourov during the Spring '11 term at University of Toronto- Toronto.

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L8 Comparing Population Means and Variances - Comparing...

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