西南交通大学：《管理统计学》（双语版） 第8章 统计估计

第8章统计估计 (StatisticaL Estimation)

第8章 统计估计 (Statistical Estimation)

基本概念 1、 Sample Statistic(样本统计量 a Any number computed from sample data Arandom variable. Known Example: Average weekly food expenditures for 100 sampled residents Random? Yes! Due to randomness of sample selection 2、 Population Parameter(总体参数) o Any number computed for the entire population A fixed number, Unknown Example: mean weekly food expenditures for all 77, 386 residents ) Do we ever know this? NO >)But we estimate it (with error)

一、基本概念 1、Sample Statistic（样本统计量）  Any number computed from sample data ⚫ A random variable. Known Example: Average weekly food expenditures for 100 sampled residents »Random? Yes! Due to randomness of sample selection 2、Population Parameter（总体参数）  Any number computed for the entire population ⚫ A fixed number. Unknown Example: mean weekly food expenditures for all 77,386 residents »Do we ever know this? NO! »But we estimate it (with error)

3、 Estimator(估计量) aA sample statistic used to guess a population parameter Example: Sample average for o selected residents is an estimator of the population mean of all77 386 residents Estimate [WRONG! Estimators are usually wrong. Often useful anyway a The actual number computed from the data Example: 33.91 is an estimate of neighborhood weekly food expenditures per person Estimation error:估计误差 a Estimator minus population parameter. Unknown o Example:

3、Estimator（估计量）  A sample statistic used to guess a population parameter ⚫ Example: Sample average for 100 selected residents is an estimator of the population mean of all 77,386 residents Estimate [WRONG! Estimators are usually wrong. Often useful anyway]  The actual number computed from the data ⚫ Example: $33.91 is an estimate of neighborhood weekly food expenditures per person Estimation error:估计误差  Estimator minus population parameter. Unknown ⚫ Example: 33.91 – 35.69 = –1.78

估计及其分布 1、计量资料 E(X)=1 x=a/√n N(L,o/n)

二、 估计及其分布 1、计量资料 ( , / ) / ( ) 2 X ~ N n S n E X n X     → = =

2、计数资料 E()=T Sp=√x(1-x)/n n→∞ N(x,丌(1-x)/n) 3、区间估计:参考值范围及可信区间(R,CI) X(P)<SD;X(P)±4SE X±1.64·S:P±1.64·S。→90%C1 X X±1.96·SP±1.96·S。→95%0C X X±2.58.S:P±2.58·S→99%C1

2、计数资料 3、区间估计：参考值范围及可信区间（RI,CI） ( , (1 )/ ) (1 )/ ( ) P ~ N n S n E P n P       − = − = → X S P S C I X S P S C I X S P S C I X P SD X P SE X P X P X P 2.58 ; 2.58 99% 1.96 ; 1.96 95% 1.64 ; 1.64 90% ( ) ; ( )                     

Estimation process 估计过程 Population Random Sample I am 95%o Mean confident that u is Mean,μ,is X=50 between 40 &60 unknown Sample

Mean, , is unknown Population Random Sample I am 95% confident that  is between 40 & 60. Mean X = 50 Estimation Process 估计过程 Sample

Population Parameters Estimated 总体参数估计 Estimate Population with Sample Parameter Statistic Mean X Proportion p Variance DifferenceμrH2 X - X

Estimate Population Parameter... with Sample Statistic Mean  Proportion p ps Variance s 2 Population Parameters Estimated 总体参数估计  2 Difference  -  1 2 x - x 1 2 X _ _ _

Confidence Interval estimation 置信区间估计 Provides range of values n Based on Observations from 1 Sample Gives Information about closeness to Unknown Population Parameter Stated in terms of Probability Never 100%o sure

• Provides Range of Values  Based on Observations from 1 Sample • Gives Information about Closeness to Unknown Population Parameter • Stated in terms of Probability Never 100% Sure Confidence Interval Estimation 置信区间估计

Elements of confidence Interval Estimation(原理) A Probability That the Population Parameter Falls Somewhere Within the Interval Confidence Interval Confidence Limit Confidence Limit (Upper)

Confidence Interval Sample Statistic Confidence Limit (Lower) Confidence Limit (Upper) A Probability That the Population Parameter Falls Somewhere Within the Interval. Elements of Confidence Interval Estimation（原理）

Confidence limits for Population Mean 总体均数的置信限 Parameter H=X± Error Statistic± Its error X-F=Error X Error X Error =20 Ⅹ±Z

Parameter = Statistic ±Its Error Confidence Limits for Population Mean 总体均数的置信限  = X  Error = Error =  − X X X X Z    = − = = Z  x  X Z X =  Error Error X − 