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Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics for Economist Ch. 18 Accuracy of Average 1. Sampling Distribution and Standard Error 2. Sample Average 3. What kind of Standard Error to be used? 4. Remember This 5. Measurement Error Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics INDEX STATISTICS 1 Sampling Distribution and Standard Error 2 Sample Average 3 What kind of Standard Error to be used? 4 Remember This 5 Measurement Error Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 2/21 STATISTICS 1. Sampling Distribution and Standard Error Accuracy of Sample Average How big the difference will be between Random Sample Average from the Black box and Population Average? Below is a result of random replacement sampling (25 times) drawn out from a box containing cards that have number from 1 to 7 on each other . And This process has done twice (Sample size 25) 2 4 3 2 5 7 5 6 4 5 4 4 1 2 4 4 6 4 7 2 7 2 Sample Sum = 105 573 Sample Average =105/25=4.2 51434 52177 12324 71653 66 Sample Sum = 95 334 Sample Average =94/25=3.8 At Random Sampling Accident 25 Numbers gonna be changed after another sampling Expect Value of Sample This would and Standard Error result in Another Sample indicate the confidence of Estimates. Average Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 3/21 1.Sampling Distribution and Standard Error STATISTICS Sample Average of Random replacement Sampling S.D of box : When One sample got drawn out, S.D indicates the deviation from the population average of the drawn value. S.E. of Sample Average : This indicates the deviation from the population average of Drawn Sample average. S.D of Sample : This indicates the deviation of drawn one Sample from the sample average. That is, This is the estimate of S.D of Box Expect Value of S.A. = Average of Box S.E. of S.A. = S.E. of Sample Sum/Sample Size = S.D of Box()/ S.D. of Sample(SD)/ Sample Size Sample Size Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 4/21 STATISTICS 1. Sampling Distribution and Standard Error Sample Average of Random replacement Sampling Ex1) At the 25 times random replacement sampling drawn out from a box containing cards that have number from 1 to 7 on each other, Calculate the Expect Value of Sample Average and Standard Error. Average of Box = 4 Expect Value of Sample Sum = 425 = 100 S.D of Box = 2 S.E of Sample Sum = 2 25 = 10 Sample Sum: 10010 Sample Average: 40.4 (S.A is the value that is Sample sum to sample size(25)) Expect value of S.A. = 4, S.E.= 0.4 Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 5/21 1. Sampling Distribution and Standard Error STATISTICS Sampling Distribution Probability Histogram of Sample Average (Sampling Distribution) : As Restoring Drawing Sample Average infinitely, We can get histogram of sample average consist of infinite sample averages. 표준단위 -3 -2 -1 0 표준단위 1 2 3 50 -3 -2 -1 0 1 2 3 2.8 3.2 3.6 4 4.4 4.8 5.2 표준단위별 비율 (%) 표준단위별 비율 (%) 50 25 25 0 0 70 80 90 100 110 120 130 합 평균 - This Two Histograms have difference only in scale, over whole shape is identical. - This Two Histogram approximate to Normal distribution asymptotically by Central Limit Theorem. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 6/21 1. Sampling Distribution and Standard Error STATISTICS Sampling Distribution Ex2) In Ex1)What is Expect value of Sample Average and Standard Error, if We draw out sample 100 time instead of 25 times. Average of Box = 4 Expect Value of Sample Sum = 4100 = 400 S.D of Box = 2 Sample Sum : 400 20 S.E of Sample Sum = 2 100 = 20 Sample Average : 4 0.2 S.E. of Sample Sum=20(increased), S.E of Sample Average=0.2(decreased) As S.E of Sample Sum is in proportional to Square root of Sample size, Sample Average-Sample sum/sample size- is in inverse proportional to Square root of Sample size. That is S.E. of Sample average got decreased Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 7/21 INDEX STATISTICS 1 Sampling Distribution and Standard Error 2 Sample Average 3 What kind of Standard Error to be used? 4 Remember This 5 Measurement Error Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 8/21 2. Sample Average STATISTICS Inference Inference: How to know Population Average from Sample Average if We don’t have any information of Box? Ex) We got random sample of 1,000 households from The City has 20,500 households, and Sample Average of Household income is KRW32.4mil.. What is the Average Household income of Population? S.D of Drawn 1,000 households=KRW19mil. S.E. of Sample Sum=KRW19mil. 1000 =KRW600mil. S.E of Sample Average=KRW600mil./1000=KRW0.6mil. Average Income of Whole Households is inferred to KRW32.4mil. KRW0.6mil. ‘Observed Value of Sample Average= Average of Box + Probable Error’, ‘Probability Error S.E of Sample Average’ We Can infer Average of Box from Observed value of Sample Average Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 9/21 2. Sample Average STATISTICS Confidence Interval Confidence Level 95%: 95% of All confidence intervals include Population Average 95% Confidence Interval of Average Households: Sample Average +/- 2*S.E. income of 25,000 KRW32.4mil.1.2mil. (KRW31.2mil., KRW33.6mil.) Why S.E. is used instead of S.D.? S.D.: When One sample got drawn out, S.D indicates the deviation from the population average. S.E. of Sample Average: This indicates the deviation from the population average of Drawn Sample average. If We want to get Population Average from Sample Average, We should use S.E. of Sample Average Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 10/21 2. Sample Average STATISTICS Confidence Interval 95% confidence interval covers area - 2~+2 in Standard Normal Distribution Curve Why Standard Normal Distribution Curve is used in calculating confidence interval? Central Limit Theorem: If A histogram of individual observed value is not identical to Normal Distribution Curve, Shape of Probability Histogram of Sample Average approximates to Normal Distribution Curve. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 11/21 2. Sample Average STATISTICS Educational Period of Population aged 25 : Distribution of Sample resembles Population Distribution. Why? The Law of Large Number – Empirical Histogram approximates to Probability Histogram If Sample size were enough, Sample Average infers Population Average and Sample Standard Deviation infer Population Standard Deviation 25세 인구 전체 (모집단) 400명을 무작위추출 (표본) 표본평균의 분포 Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 12/21 2. Sample Average STATISTICS Sampling Distribution There is great difference between Distribution of Population and Normal Distribution. But Probability Histogram of Sample Average (Sample Distribution of Sample Average) resembles Normal Distribution Curve In Simple Random Sample of Sized 400 people, What is the probability of event that Sample Average is in 11.2(year) and 12(year) = Area of (-2, 2) section in Standard Normal Distribution Curve = 95% 95% confidence interval of Average Education Period of Population = (11.2(year), 12.0(year)) A individual observed value does not follow Normal Distribution Curve, Shape of Probability Histogram of Sample Average resembles to Normal Distribution Curve. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 13/21 INDEX STATISTICS 1 Sampling Distribution and Standard Error 2 Sample Average 3 What kind of Standard Error to be used? 4 Remember This 5 Measurement Error Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 14/21 3. What kind of Standard Error to be used? STATISTICS S.E in Box Model Inferring Sum S.E. of Sample Sum Inferring Average S.E. of Sample Average Inferring Number S.E. of Sample Number Inferring Ration S.E. of Sample Ratio S.E. of Sample Sum = S.D. of Box× S.E. of Sample Average = S.D. of Box/ Sample Size Sample Size S.E. of Sample Size = S.E. of 0-1 Box Sample Sum S.E. of Sample Ratio = (S.E. of Sample Number/Sample Size)×100% In General, We don’t know what the S.D. of Box is, We replace S.D. of Box to known Sample S.D. to calculate S.E. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 15/21 INDEX STATISTICS 1 Sampling Distribution and Standard Error 2 Sample Average 3 What kind of Standard Error to be used? 4 Remember This 5 Measurement Error Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 16/21 STATISTICS 4. Remember This Box Model There is difference between Sample Average and Population Average and that is Probable Error Less Probable error means More confidence level, More Probable Error means Less confidence level. General Size of Probable Error is inferred from S.E. S.E of Sample Average is inferred by dividing Sample S.E by Square root of Sample size. Random Sampling let us calculate S.E. of Sample Average directly from only Sample size and Sample S.D. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 17/21 STATISTICS 4. Remember This Statistical Inference Statistical Inference? Given Data Stochastic Method Characteristic of Population Many Method learned until now is meaningless in model that is not conversed to Box model. To do Statistical inference, We need a stochastic model like a box model. Statistical Inference is based on Stochastic Method. It is meaningless to calculate S.E. in a model that is not conversed to the Box model (ex. Including trend or intention). Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 18/21 INDEX STATISTICS 1 Sampling Distribution and Standard Error 2 Sample Average 3 What Standard Error to be used? 4 Remember This 5 Measurement Error Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 19/21 STATISTICS 5. Measurement Error Measurement Error Measurement Error : Not in Object itself, There are Stochastic Errors in Measuring process, A Observed Value has measurement error compared to the Real Value. Repeating more and more and then Averaging reduces a Measurement Error (inverse proportional to square root of number of repetition), increases 정 Confidence of inference (proportional to square root of repetition). Ex) 100 times repetition of measuring weight of 1 gobbet of meat Average = 600g, Standard Deviation=10g •Size of Measurement Error contained in 1 observation: About 10g (Sample Standard Deviation) •Size of Measurement Error contained in Observed Average : About 1g(=10g/ 100 )(S.E. of Sample Average) 95% Confidence Interval of Real Value of weight (600-2)g (600+2)g Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 20/21 5. Measurement Error STATISTICS Gaussian Model Gaussian Model Box Model Drawing out many cards from one box Repeating observation for identical object in same condition Gaussian Model: Measurement Error is identical to drawing out a error card in a single trial from a box Ex) 1st Observation = Real Value+1st drawing out from a error box 2nd Observation = Real Value+2nd drawing out from a error box …… : 100th Observation = Real Value+100th drawing out from a error box S.D. of observations of repetition = S.D. of Probable Errors In Gaussian Model, S.D. of Observations in repetition is a estimate of S.D. of a Error box. As Sample size get larger, Confidence level of a Estimate increases. Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics Statistics & Econometrics 21/21