Chapter 9: Elementary Sampling Theory

Transcription

1 Chapter 9: Elementary Sampling Theory James B. Ramsey Economics; NYU Ramsey (Institute) Chapter 9: / 20

2 Sampling Theory is the LINK between Theory & Observation Chapters 1 to 5: Data Descriptions Chapters 6 to 8: Development of Probability Theory Chapter 9 to 11; the Basics of Inference Chapter 12 to 14: Introduction to Regression Chapter 15: Retrospective-the use & interpretation of data & news Ramsey (Institute) Chapter 9: / 20

3 What do we want to Learn? 1 What Probability Distribution applies? Ramsey (Institute) Chapter 9: / 20

4 What do we want to Learn? 1 What Probability Distribution applies? 2 What is the de ning statistical experiment? Ramsey (Institute) Chapter 9: / 20

5 What do we want to Learn? 1 What Probability Distribution applies? 2 What is the de ning statistical experiment? 3 How to discover the relevant parameter values? Ramsey (Institute) Chapter 9: / 20

6 What do we want to Learn? 1 What Probability Distribution applies? 2 What is the de ning statistical experiment? 3 How to discover the relevant parameter values? 4 How do we infer the properties of the experiment in future trials? Ramsey (Institute) Chapter 9: / 20

7 What do we want to Learn? 1 What Probability Distribution applies? 2 What is the de ning statistical experiment? 3 How to discover the relevant parameter values? 4 How do we infer the properties of the experiment in future trials? 5 What procedures do we use to link theory and data? Ramsey (Institute) Chapter 9: / 20

8 What do we want to Learn? 1 What Probability Distribution applies? 2 What is the de ning statistical experiment? 3 How to discover the relevant parameter values? 4 How do we infer the properties of the experiment in future trials? 5 What procedures do we use to link theory and data? 6 We will answer item three and ve rst as they are the easiest. Ramsey (Institute) Chapter 9: / 20

9 Example: Book publishers and typographical errors. The number of errors in a block of 10 pages Suppose the relevant distribution is Poisson and λ = 2. Take a sample of 10 pages and count the number of typos. This is one observation, say 3 or 4 or 6 even if λ = 2. If we did not know λ, what would we conclude? Ramsey (Institute) Chapter 9: / 20

10 Question: What if we averaged a group of n independent observations, x 1,x 2,......x n? Consider: X= 1 n X i E f X g = λ Ramsey (Institute) Chapter 9: / 20

11 Question: What if we averaged a group of n independent observations, x 1,x 2,......x n? Consider: X= 1 n X i But what of µ 2 ( X )? E f X g = λ Ramsey (Institute) Chapter 9: / 20

12 Question: What if we averaged a group of n independent observations, x 1,x 2,......x n? Consider: X= 1 n X i But what of µ 2 ( X )? E f X g = λ Recall: For any variable: µ 2 ( X ) = µ 0 2 ( X ) µ 0 1 ( X ) 2 Ramsey (Institute) Chapter 9: / 20

13 Question: What if we averaged a group of n independent observations, x 1,x 2,......x n? Consider: X= 1 n X i But what of µ 2 ( X )? E f X g = λ Recall: For any variable: µ 2 ( X ) = µ 0 2 ( X ) µ 0 1 ( X ) 2 For our Poisson Distn: µ 0 2 ( X ) = 1 E n n( X ) 2o 2 Ramsey (Institute) Chapter 9: / 20

14 Question: What if we averaged a group of n independent observations, x 1,x 2,......x n? Consider: X= 1 n X i But what of µ 2 ( X )? E f X g = λ Recall: For any variable: µ 2 ( X ) = µ 0 2 ( X ) µ 0 1 ( X ) 2 For our Poisson Distn: µ 0 2 ( X ) = 1 E n n( X ) 2o 2 = 1 n 2 E X 2 + i6=j X i X j Ramsey (Institute) Chapter 9: / 20

15 Question: What if we averaged a group of n independent observations, x 1,x 2,......x n? Consider: X= 1 n X i But what of µ 2 ( X )? E f X g = λ Recall: For any variable: µ 2 ( X ) = µ 0 2 ( X ) µ 0 1 ( X ) 2 For our Poisson Distn: µ 0 2 ( X ) = 1 E n n( X ) 2o 2 = 1 n 2 E X 2 + i6=j X i X j = 1 n 2 λ + λ 2 + i6=j λ 2 Ramsey (Institute) Chapter 9: / 20

16 Question: What if we averaged a group of n independent observations, x 1,x 2,......x n? Consider: X= 1 n X i But what of µ 2 ( X )? E f X g = λ Recall: For any variable: µ 2 ( X ) = µ 0 2 ( X ) µ 0 1 ( X ) 2 For our Poisson Distn: µ 0 2 ( X ) = 1 E n n( X ) 2o 2 = 1 E n 2 X 2 + i6=j X i X j = 1 n λ + λ i6=j λ 2 λ + λ 2 + (n 1) λ 2 = 1 n Ramsey (Institute) Chapter 9: / 20

17 Question: What if we averaged a group of n independent observations, x 1,x 2,......x n? Consider: X= 1 n X i But what of µ 2 ( X )? E f X g = λ Recall: For any variable: µ 2 ( X ) = µ 0 2 ( X ) µ 0 1 ( X ) 2 For our Poisson Distn: µ 0 2 ( X ) = 1 E n n( X ) 2o 2 = 1 E n 2 X 2 + i6=j X i X j = 1 n λ + λ i6=j λ 2 = 1 n λ + λ 2 + (n 1) λ 2 or µ 2 ( X ) = 1 n λ + λ 2 + (n 1) λ 2 λ 2 Ramsey (Institute) Chapter 9: / 20

18 Question: What if we averaged a group of n independent observations, x 1,x 2,......x n? Consider: X= 1 n X i But what of µ 2 ( X )? E f X g = λ Recall: For any variable: µ 2 ( X ) = µ 0 2 ( X ) µ 0 1 ( X ) 2 For our Poisson Distn: µ 0 2 ( X ) = 1 E n n( X ) 2o 2 = 1 E n 2 X 2 + i6=j X i X j = 1 n λ + λ i6=j λ 2 = 1 n λ + λ 2 + (n 1) λ 2 or µ 2 ( X ) = 1 n λ + λ 2 + (n 1) λ 2 λ 2 = λ n Ramsey (Institute) Chapter 9: / 20

19 From the variance of X, just calculated as µ 2 ( X ) = λ n. For n large enough, We can claim: X is almost certainly equal to the unknown λ. For example: λ = 2, n = 10,000; variance of X is = The corresponding standard deviation is See graphs on overhead. Ramsey (Institute) Chapter 9: / 20

20 Sampling Theory: the Basics The objectives of sample design are to ensure: 1 Drawings are independent Ramsey (Institute) Chapter 9: / 20

21 Sampling Theory: the Basics The objectives of sample design are to ensure: 1 Drawings are independent 2 Each drawing is from the postulated distribution Ramsey (Institute) Chapter 9: / 20

22 Sampling Theory: the Basics The objectives of sample design are to ensure: 1 Drawings are independent 2 Each drawing is from the postulated distribution 3 The process of sampling does not alter the parent distribution Ramsey (Institute) Chapter 9: / 20

23 Above requirements can be summarized as follows: If G(Xjθ) is the assumed parent distribution with parameters θ; Ramsey (Institute) Chapter 9: / 20

24 Above requirements can be summarized as follows: If G(Xjθ) is the assumed parent distribution with parameters θ; The joint distribution of the sample given the conditions of the experiment is: Ramsey (Institute) Chapter 9: / 20

25 Above requirements can be summarized as follows: If G(Xjθ) is the assumed parent distribution with parameters θ; The joint distribution of the sample given the conditions of the experiment is: F j (X 1, X 2, X 3,...X n jc.e.) = Π i G(X i jθ) Ramsey (Institute) Chapter 9: / 20

26 Above requirements can be summarized as follows: If G(Xjθ) is the assumed parent distribution with parameters θ; The joint distribution of the sample given the conditions of the experiment is: F j (X 1, X 2, X 3,...X n jc.e.) = Π i G(X i jθ) This structure enables us to: Ramsey (Institute) Chapter 9: / 20

27 Above requirements can be summarized as follows: If G(Xjθ) is the assumed parent distribution with parameters θ; The joint distribution of the sample given the conditions of the experiment is: F j (X 1, X 2, X 3,...X n jc.e.) = Π i G(X i jθ) This structure enables us to: 1 Draw general conclusions from speci c observations; Ramsey (Institute) Chapter 9: / 20

28 Above requirements can be summarized as follows: If G(Xjθ) is the assumed parent distribution with parameters θ; The joint distribution of the sample given the conditions of the experiment is: F j (X 1, X 2, X 3,...X n jc.e.) = Π i G(X i jθ) This structure enables us to: 1 Draw general conclusions from speci c observations; 2 Infer the values of parameters & moments from actual observations; Ramsey (Institute) Chapter 9: / 20

29 Above requirements can be summarized as follows: If G(Xjθ) is the assumed parent distribution with parameters θ; The joint distribution of the sample given the conditions of the experiment is: F j (X 1, X 2, X 3,...X n jc.e.) = Π i G(X i jθ) This structure enables us to: 1 Draw general conclusions from speci c observations; 2 Infer the values of parameters & moments from actual observations; 3 Development of the sampling theory aids interpretation of observations Ramsey (Institute) Chapter 9: / 20

30 "Population" and Sampling Sampling is the link between probability theory & observations; "Drawings" are from the "hypothetical" parent distribution; Do not confuse this with physical drawings from a physical "population:" An old idea: Compare sampling sh from a pond: Do you want to calculate distribution of this pond today? Or do you want to INFER properties of comparable ponds in the future? The choice is between DESCRIPTION and INFERENCE. We are now interested in INFERENCE. If a "physical population", cf. sh in a pond,sampling without replacement alters the population; not so with a hypothetical population! Ramsey (Institute) Chapter 9: / 20

31 "Population" and Sampling: continued At any point in time the sh in the pond can be regarded as a "sample" from the hypothetical population of sh in the pond. At all times sh are constantly dying, being born; the physical characteristics are changing! Distinguish: Simple random sampling from strati ed random sampling which is used to sample from populations composed of sub-groups. Ramsey (Institute) Chapter 9: / 20

32 Examples of the Dangers of Errors in Sample Design Landon/Roosevelt election: use of car and phone registrations instead of voter registrations. Ramsey (Institute) Chapter 9: / 20

33 Examples of the Dangers of Errors in Sample Design Landon/Roosevelt election: use of car and phone registrations instead of voter registrations. Process of sampling alters parent distribution: sh pond again! Ramsey (Institute) Chapter 9: / 20

34 Examples of the Dangers of Errors in Sample Design Landon/Roosevelt election: use of car and phone registrations instead of voter registrations. Process of sampling alters parent distribution: sh pond again! National Law Journal poll: What ever the judge says the law is, jurors should do what they think is the right thing; 3/4 agree!. Ramsey (Institute) Chapter 9: / 20

35 Examples of the Dangers of Errors in Sample Design Landon/Roosevelt election: use of car and phone registrations instead of voter registrations. Process of sampling alters parent distribution: sh pond again! National Law Journal poll: What ever the judge says the law is, jurors should do what they think is the right thing; 3/4 agree!. Versus: Regardless of how jurors personally view the case, they should always follow the instructions of the judge concerning the law in the case; 3/4 agree! Ramsey (Institute) Chapter 9: / 20

36 Results of a survey depend on who asks & how: cf. Nth. Carolina survey in an experiment; Ramsey (Institute) Chapter 9: / 20

37 Results of a survey depend on who asks & how: cf. Nth. Carolina survey in an experiment; Desire to please; make believe T.V. shows; Ramsey (Institute) Chapter 9: / 20

38 Results of a survey depend on who asks & how: cf. Nth. Carolina survey in an experiment; Desire to please; make believe T.V. shows; Self -selection bias in surveys; Ramsey (Institute) Chapter 9: / 20

39 Results of a survey depend on who asks & how: cf. Nth. Carolina survey in an experiment; Desire to please; make believe T.V. shows; Self -selection bias in surveys; Self-image & desire not to standout; e.g. "support" for a dictator. Ramsey (Institute) Chapter 9: / 20

40 Introduction to Proportionate Strati ed Random Sampling See overheads for sample distributions of income for social scientists; economists & sociologists. Ramsey (Institute) Chapter 9: / 20

41 Introduction to Proportionate Strati ed Random Sampling See overheads for sample distributions of income for social scientists; economists & sociologists. Note that both means and range of sub-groups di er substantially. Ramsey (Institute) Chapter 9: / 20

42 Introduction to Proportionate Strati ed Random Sampling See overheads for sample distributions of income for social scientists; economists & sociologists. Note that both means and range of sub-groups di er substantially. Interested in the joint mean for social scientists and in our estimate s spread. Ramsey (Institute) Chapter 9: / 20

43 Introduction to Proportionate Strati ed Random Sampling See overheads for sample distributions of income for social scientists; economists & sociologists. Note that both means and range of sub-groups di er substantially. Interested in the joint mean for social scientists and in our estimate s spread. De ne the joint mean, µ J, by: Ramsey (Institute) Chapter 9: / 20

44 Introduction to Proportionate Strati ed Random Sampling See overheads for sample distributions of income for social scientists; economists & sociologists. Note that both means and range of sub-groups di er substantially. Interested in the joint mean for social scientists and in our estimate s spread. De ne the joint mean, µ J, by: µ J = N 1 N µ 1 + N 2 N µ 2 N = N 1 + N 2 N 1 = No. of Economists N 2 = No. of Sociologists µ 1 = Mean for Economists µ 2 = Mean for Sociologists Ramsey (Institute) Chapter 9: / 20

45 Calculating the Joint Variance σ 2 g = E f(x µ J ) 2 g = N 1 N E 1f(X 1 µ J ) 2 g + N 2 N E 2f(X 2 = N 1 N E 1f[(X 1 µ 1 ) + (µ 1 µ J )] 2 g + N 2 N E 2f[(X 2 µ 1 ) + (µ 2 µ J )] 2 g µ J ) 2 g = N 1 N [σ2 1 + (µ 1 µ J ) 2 ] + N 2 N [σ2 2 + (µ 2 µ J ) 2 ] Ramsey (Institute) Chapter 9: / 20

46 Calculating the Joint Variance σ 2 g = E f(x µ J ) 2 g = N 1 N E 1f(X 1 µ J ) 2 g + N 2 N E 2f(X 2 = N 1 N E 1f[(X 1 µ 1 ) + (µ 1 µ J )] 2 g + N 2 N E 2f[(X 2 µ 1 ) + (µ 2 µ J )] 2 g µ J ) 2 g = N 1 N [σ2 1 + (µ 1 µ J ) 2 ] + N 2 N [σ2 2 + (µ 2 µ J ) 2 ] Given that the sub-samples are distributed independently. Ramsey (Institute) Chapter 9: / 20

47 Calculating the Joint Variance σ 2 g = E f(x µ J ) 2 g = N 1 N E 1f(X 1 µ J ) 2 g + N 2 N E 2f(X 2 = N 1 N E 1f[(X 1 µ 1 ) + (µ 1 µ J )] 2 g + N 2 N E 2f[(X 2 µ 1 ) + (µ 2 µ J )] 2 g µ J ) 2 g = N 1 N [σ2 1 + (µ 1 µ J ) 2 ] + N 2 N [σ2 2 + (µ 2 µ J ) 2 ] Given that the sub-samples are distributed independently. The overall variance is composed of four terms; 2 own variances, Ramsey (Institute) Chapter 9: / 20

48 Calculating the Joint Variance σ 2 g = E f(x µ J ) 2 g = N 1 N E 1f(X 1 µ J ) 2 g + N 2 N E 2f(X 2 = N 1 N E 1f[(X 1 µ 1 ) + (µ 1 µ J )] 2 g + N 2 N E 2f[(X 2 µ 1 ) + (µ 2 µ J )] 2 g µ J ) 2 g = N 1 N [σ2 1 + (µ 1 µ J ) 2 ] + N 2 N [σ2 2 + (µ 2 µ J ) 2 ] Given that the sub-samples are distributed independently. The overall variance is composed of four terms; 2 own variances, & the squares of the di erences between individual group means & the overall mean. Ramsey (Institute) Chapter 9: / 20

49 A} Estimators based on Simple Random Sampling; x g = 1 n x i = $70, 741 E f x g g = N 1 N µ 1 + N 2 N µ 2 = µ J Var( x g ) = E f( x g µ J ) 2 g = 1 n σ2 g ˆV ar( x g ) q = $456, 693 ˆV ar( x g ) = $676 Ramsey (Institute) Chapter 9: / 20

50 B} Estimators based on Proportionate Strati ed Sampling Let the sample sizes be n 1 & n 2, n 1 + n 2 = n, where: Ramsey (Institute) Chapter 9: / 20

51 B} Estimators based on Proportionate Strati ed Sampling Let the sample sizes be n 1 & n 2, n 1 + n 2 = n, where: n 1 = N 1 N n n 2 = N 2 N n n 1 = N 1 n 2 N 2 Ramsey (Institute) Chapter 9: / 20

52 B} Estimators based on Proportionate Strati ed Sampling Let the sample sizes be n 1 & n 2, n 1 + n 2 = n, where: n 1 = N 1 N n n 2 = N 2 N n n 1 = N 1 n 2 N 2 The corresponding estimator based on strati ed sampling is given by: x p Ramsey (Institute) Chapter 9: / 20

53 B} Estimators based on Proportionate Strati ed Sampling Let the sample sizes be n 1 & n 2, n 1 + n 2 = n, where: n 1 = N 1 N n n 2 = N 2 N n n 1 = N 1 n 2 N 2 The corresponding estimator based on strati ed sampling is given by: x p x p = N 1 N x 1 + N 2 N x 2 E f x p g = µ J Ramsey (Institute) Chapter 9: / 20

54 Variance Estimator based on Strati ed Sampling Var( x p ) = Var( N 1 N x 1 + N 2 N x 2) 2 2 N1 N2 = Var( x 1 ) + Var( x 2 ) N N 2 N1 σ = N2 σ N n 1 N n 2 Substituting: n i = N i n; yields N 1 n [ N1 σ N ˆV ar( x p ) q = $130, 281 ˆV ar( x p ) = $361 N2 N σ 2 2] Ramsey (Institute) Chapter 9: / 20

55 Sample Standard deviation under simple random sampling is: $676 Ramsey (Institute) Chapter 9: / 20

56 Sample Standard deviation under simple random sampling is: $676 Sample Standard deviation under strati ed random sampling is: $361 Ramsey (Institute) Chapter 9: / 20

57 Survey Questionnaire Experiment. EXPERIMENT in Teams: Troop levels in Iraq; Design a survey: To elicit an increase; To elicit a decrease; To be neutral. U.S. sanctions on Iran: Design a survey: To elicit an increase; To elicit a decrease; To be neutral. Minimum wage; Design a survey: To elicit an increase; To elicit a decrease; To be neutral. Present at next class. Ramsey (Institute) Chapter 9: / 20

58 End of Chapter 9 Ramsey (Institute) Chapter 9: / 20