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Lecture 09-sampling estimation - 1036 Probability...

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Prob. & Stat. Lecture09 - one-/two-sample estimation ([email protected]) 9-1 1036: Probability & Statistics 1036: Probability & 1036: Probability & Statistics Statistics Lecture 9 Lecture 9 One One - - and Two - - Sample Sample Estimation Problems Estimation Problems
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Prob. & Stat. Lecture09 - one-/two-sample estimation ([email protected]) 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
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Prob. & Stat. Lecture09 - one-/two-sample estimation ([email protected]) 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… Unbiased estimator 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
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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
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Prob. & Stat. Lecture09 - one-/two-sample estimation ([email protected]) 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
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Prob. & Stat. Lecture09 - one-/two-sample estimation ([email protected]) 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 < <
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Prob. & Stat. Lecture09 - one-/two-sample estimation ([email protected]) 9-7 Interpretation of Interval Estimates If the confidence interval is The probability that the mean is within the range can be stated as: We would state that there is a (1- α ) × 100% confidence interval of Ideally, predict narrow range with high degree of confidence U L θ θ θ ˆ ˆ < < α θ θ θ = < < 1 ) ˆ ˆ ( U L P U L θ θ θ ˆ ˆ < <
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