Math 203_Lecture 9

Math 203_Lecture 9 - Principles of Statistics 1 Lecture 9...

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Unformatted text preview: Principles of Statistics 1 Lecture 9: Math 203 Abbas Khalili Department of Mathematics and Statistics McGill University May 16, 2011 Principles of Statistics 1 Lecture 9: Math 203 – p. 1/4 3 The most common parameter of interest The population mean (average) μ is the most popular parameter that we are often interested in. Principles of Statistics 1 Lecture 9: Math 203 – p. 2/4 3 The most common parameter of interest The population mean (average) μ is the most popular parameter that we are often interested in. To estimate μ , we use the sample data. Principles of Statistics 1 Lecture 9: Math 203 – p. 2/4 3 The most common parameter of interest The population mean (average) μ is the most popular parameter that we are often interested in. To estimate μ , we use the sample data. Consider the random sample: X 1 ,X 2 ,... ,X n Principles of Statistics 1 Lecture 9: Math 203 – p. 2/4 3 The most common parameter of interest The population mean (average) μ is the most popular parameter that we are often interested in. To estimate μ , we use the sample data. Consider the random sample: X 1 ,X 2 ,... ,X n The most obvious sample statistic to use for estimating the population mean μ is the sample mean: ¯ X , ¯ X = 1 n n summationdisplay i =1 X i Principles of Statistics 1 Lecture 9: Math 203 – p. 2/4 3 Expectation and variance of ¯ X Assume E ( X i ) = μ and V ar ( X i ) = σ 2 for i = 1 , 2 ,... ,n . Principles of Statistics 1 Lecture 9: Math 203 – p. 3/4 3 Expectation and variance of ¯ X Assume E ( X i ) = μ and V ar ( X i ) = σ 2 for i = 1 , 2 ,... ,n . By simple algebra: E ( ¯ X ) = μ. Principles of Statistics 1 Lecture 9: Math 203 – p. 3/4 3 Expectation and variance of ¯ X Assume E ( X i ) = μ and V ar ( X i ) = σ 2 for i = 1 , 2 ,... ,n . By simple algebra: E ( ¯ X ) = μ. we say the sample mean ¯ X is an unbiased estimator of the population mean μ . Principles of Statistics 1 Lecture 9: Math 203 – p. 3/4 3 Expectation and variance of ¯ X Assume E ( X i ) = μ and V ar ( X i ) = σ 2 for i = 1 , 2 ,... ,n . By simple algebra: E ( ¯ X ) = μ. we say the sample mean ¯ X is an unbiased estimator of the population mean μ . Also, V ar ( ¯ X ) = σ 2 n . Principles of Statistics 1 Lecture 9: Math 203 – p. 3/4 3 CENTRAL LIMIT THEOREM (CLT) We have established then that the sample mean ¯ X is a very good estimator of μ , because of E ( ¯ X ) = μ . Also for large samples the variance of ¯ X around μ will be very small. Principles of Statistics 1 Lecture 9: Math 203 – p. 4/4 3 CENTRAL LIMIT THEOREM (CLT) We have established then that the sample mean ¯ X is a very good estimator of μ , because of E ( ¯ X ) = μ . Also for large samples the variance of ¯ X around μ will be very small....
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This note was uploaded on 07/15/2011 for the course MATH 203 taught by Professor Dr.josecorrea during the Summer '08 term at McGill.

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Math 203_Lecture 9 - Principles of Statistics 1 Lecture 9...

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