Chapter 9 - Chapter 9 One and Two-Sample Estimation...

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Unformatted text preview: Chapter 9. One- and Two-Sample Estimation Problems 9.1 Statistical Inference ▪ using sample data, we search the right value of unknown parameter. ▪ parameter : constants which determine the distribution. ▪ estimation : estimator ▪ hypothesis testing : reject H or not Population Sample ▲ ○ × ▲ × × × × ○ ○ ○ ○ ○ ▲ ▲ ▲ × ▲ ▲ ▲ × × ○ ○ Parameters Parameters 2 , σ µ Statistics Statistics 2 , x s Statistical procedure Inference Sampling technique Population Sample ▲ ○ × ▲ × × × × ○ ○ ○ ○ ○ ▲ ▲ ▲ × ▲ ▲ ○ × ▲ × × × × ○ ○ ○ ○ ○ ▲ ▲ ▲ × ▲ ▲ ▲ × × ○ ○ ▲ ▲ × × ○ ○ Parameters Parameters 2 , σ µ Statistics Statistics 2 , x s Statistical procedure Inference Sampling technique 9.2 Classical Methods of Estimation ▪ Point estimation ▪ Interval estimation ▪ Estimator : a random variable, a procedure to estimating a parameter using sample statistics ▪ Estimate : estimated value of a parameter, not a random variable ▪ parameter : θ , estimator : ˆ θ ▪ Unbiasedness : A statistic ˆ θ is said to be an unbiased estimator of the parameter θ if ˆ ˆ ( ) E θ µ θ θ = = ▪ Mean Squared Error (MSE) : If we consider all possible unbiased estimators of some parameter θ , the one with the smallest variance is called the most efficient estimator of θ 2 2 ˆ ˆ ˆ ˆ [ ] [( ) ] [ ] [ ( ) ] MSE E Var E θ θ θ θ θ θ = − = + − If ˆ θ is an unbiased estimator of θ ˆ ˆ [ ] [ ] MSE Var θ θ = ▪ Interval Estimation : interval ˆ ˆ [ , ] L U θ θ is (1 )100% α − confidence interval for θ ˆ ˆ [ ] 1 L U P θ θ θ α < < = − 9.3 Single Sample : Estimating the Mean 9....
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Chapter 9 - Chapter 9 One and Two-Sample Estimation...

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