Lecture 09-sampling estimation

Lecture 09-sampling estimation - Prob. & Stat....

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Unformatted text preview: Prob. & Stat. Lecture09 - one-/two-sample estimation (cwliu@twins.ee.nctu.edu.tw) 9-1 1036: Probability & Statistics 1036: Probability & 1036: Probability & Statistics Statistics Lecture 9 Lecture 9 – – One One-- and Two and Two-- Sample Sample Estimation Problems Estimation Problems Prob. & Stat. Lecture09 - one-/two-sample estimation (cwliu@twins.ee.nctu.edu.tw) 9-2 Statistical Inference • Estimation – to estimate the population parameters – Classical • Based on random sample – Bayesian • Based on prior subjective knowledge about the prob. distribution and random sample • Tests of hypothesis – an assertion or conjecture concerning one or two populations Prob. & Stat. Lecture09 - one-/two-sample estimation (cwliu@twins.ee.nctu.edu.tw) 9-3 Unbiased Estimator • An estimator may not expect to estimate the exact value of population parameter – But hope that it is not fall off… • U n b i a s e d e s t i m a t o r – A statistic is said to be unbiased if its sampling distribution has a mean equal to the parameter estimated θ µ θ = Θ = ) ˆ ( E Prob. & Stat. Lecture09 - one-/two-sample estimation (cwliu@twins.ee.nctu.edu.tw) 9-4 Show that S 2 is an unbiased estimator of the parameter σ 2 2 2 1 2 2 2 2 2 2 2 1 2 2 1 2 2 1 2 1 2 2 1 2 1 1 ) ( Therefore, and , , 2 , 1 for However, 1 1 ) ( ) ( 1 1 ) ( ) ( 1 1 )] ( ) [( 1 1 1 ) ( ) ( σ σ σ σ σ σ σ σ σ µ µ µ µ µ µ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ − − = = = = ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − − − = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − − = ⎥ ⎦ ⎤ ⎢ ⎣ ⎡ − − − − = ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ − − = ∑ ∑ ∑ ∑ ∑ ∑ = = = = = = n n n S E n n i n n X nE X E n X n X E n X X E n n X X E S E n i X X X n i X n i i n i i n i i n i i i i K Prob. & Stat. Lecture09 - one-/two-sample estimation (cwliu@twins.ee.nctu.edu.tw) 9-5 Most Efficient Estimator • The one with the smallest variance among all possible unbiased estimators of some population θ is called the most efficient estimator of θ . unbiased Prob. & Stat. Lecture09 - one-/two-sample estimation (cwliu@twins.ee.nctu.edu.tw) 9-6 Interval Estimate • Even most efficient unbiased estimator not likely to estimate exactly correctly. • Although accuracy increases with large samples, no reason why a point estimate from a sample should exactly equal the population parameter. • One way to handle this error is through an interval estimate – Example: sample mean = 540 Confidence interval : 520 < m < 560 – Since s x 2 = σ 2 /n , accuracy should increase with n and interval size should decrease. – Range of interval indicates accuracy of the point estimate U L θ θ θ ˆ ˆ < < Prob. & Stat. Lecture09 - one-/two-sample estimation (cwliu@twins.ee.nctu.edu.tw) 9-7 Interpretation of Interval Estimates • If the confidence interval is • The probability that the mean is within the range can be stated as:...
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This note was uploaded on 08/23/2009 for the course IEE 1036 taught by Professor Cwliu during the Spring '06 term at National Chiao Tung University.

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Lecture 09-sampling estimation - Prob. & Stat....

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