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) 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) is the most popular parameter that we are often interested in. To estimate , 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) is the most popular parameter that we are often interested in. To estimate , 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) is the most popular parameter that we are often interested in. To estimate , 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 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 ) = and V ar ( X i ) = 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 ) = and V ar ( X i ) = 2 for i = 1 , 2 ,... ,n . By simple algebra: E ( X ) = . Principles of Statistics 1 Lecture 9: Math 203 p. 3/4 3 Expectation and variance of X Assume E ( X i ) = and V ar ( X i ) = 2 for i = 1 , 2 ,... ,n . By simple algebra: E ( X ) = . we say the sample mean X is an unbiased estimator of the population mean . Principles of Statistics 1 Lecture 9: Math 203 p. 3/4 3 Expectation and variance of X Assume E ( X i ) = and V ar ( X i ) = 2 for i = 1 , 2 ,... ,n . By simple algebra: E ( X ) = . we say the sample mean X is an unbiased estimator of the population mean . Also, V ar ( X ) = 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 , because of E ( X ) = . Also for large samples the variance of X around 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 , because of E ( X ) = . Also for large samples the variance of X around will be very small....
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Math 203_Lecture 9 - Principles of Statistics 1 Lecture 9:...

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