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[63]
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["Only the first 100 repetitions are displayed on this worksheet."]

{"sheet":"TwoVarMCSim","index":1,"n_rows":103,"n_cols":13,"rows_kept":103,"cut":false,"hidden":true,"merged":[[1,5,1,6],[3,8,3,9],[3,10,3,11],[3,12,3,13]]}
["Results of Monte Carlo Simulation"]
["Sample Number","Var1","Var2",null,null,"Simulation Stats"]
[1,null,null,null,null,null,"repetitions"]
[2,null,null,null,null,null,"secs",null,"Var1",null,"Var2",null,"Notes"]
[3,null,null,null,null,null,null,null,"Average",3.8919870765226666E-3,"Average",3.3556206172831171E-3]
[4,null,null,null,null,null,null,null,"SD",0.16598652994176216,"SD",0.14882661448398982]
[5,null,null,null,null,null,null,null,"Max",0.50081070155017959,"Max",0.40019025312581391]
[6,null,null,null,null,null,null,null,"Min",-0.54963495662729456,"Min",-0.46734164154276719]
[7]
[8]
[9]
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[11]
[12]
[13]
[14]
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[17,null,null,null,null,1]
[18]
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["Only the first 100 repetitions are displayed on this worksheet."]

{"sheet":"Intro","index":2,"n_rows":36,"n_cols":28,"rows_kept":36,"cut":false,"hidden":false,"merged":[]}
["This workbook illustrates the effect of probability of selection on sample regression estimates"]
[]
["Source Paper:"]
["Barreto and Raghav (2013) \"Understanding and Teaching Unequal Probability of Selection\", Journal of Econometric Methods:"]
["https://www.degruyterbrill.com/document/doi/10.1515/jem-2012-0017/html"]
[]
["This workbook contains Visual Basic code for two key functions:"]
["1) SAMPLE(PopRange, ProbRange, n, Optional WithRep)  enables sampling from a population"]
["2) LINESTW(YRange, XRange, WeightRange) computes probability weighted least squares estimates   "]
[]
["You must enable macros for these functions to work."]
[]
["Both functions are Excel array functions. Instead of Enter, Ctrl-Shift-Enter is required. Click Esc to exit an array function without making any changes."]
[]
["Monte Carlo simulation code in this workbook is based on Barreto and Howland's MCSim.xla Excel add-in."]
["http://www.wabash.edu/econometrics/EconometricsBook/Basic%20Tools/ExcelAddIns/MCSim.htm"]
[]
["The Idea:",null,null,null,null,null,null,null,null,null,null,null,null,"Toy Sampling Example:"]
["A simple random sample (SRS) means each item in the population is equally likely to be chosen -- simple means basic, not easy.",null,null,null,null,null,null,null,null,null,null,null,null,"Sample size of one."]
["Much more common in surveys, especially large, complex surveys like the CPS, is unequal probability of selection.",null,null,null,null,null,null,null,null,null,null,null,null,"Hospital","Patients",null,null,null,null,null,null,null,"Hospital A","Hospital B",null,null,"1 is a particular patient is selected"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,"A",4,null,null,null,null,null,null,null,"Rolando","Jose",{"$f":"IF(RAND()<0.5,\"A\",\"B\")","v":"A"},"selected hospital",{"$f":"IF($Y$23=W21,1,0)","v":0},{"$f":"IF($Y$23=X21,1,0)","v":0}]
["Suppose you want to sample hospital patients. There is no list of hospital patients, but there is a list of hospitals.",null,null,null,null,null,null,null,null,null,null,null,null,"B",2,null,null,null,null,null,null,null,"Mariceli","Humberto",{"$f":"RAND()","v":0.97825186483781124},"RAND",{"$f":"IF($Y$23=W22,1,0)","v":0},{"$f":"IF($Y$23=X22,1,0)","v":0}]
["You decide to sample from the hospital list and then survey every patient in the hospitals in your sample.",null,null,null,null,null,null,null,null,null,null,null,null,"SRS is \"put all patients in a box, draw 1\" so each patient has a 1/6 chance of being selected.",null,null,null,null,null,null,null,null,"Jorge",null,{"$f":"IF(Y21=\"A\",IF(Y22<0.25,W21,IF(Y22<0.5,W22,IF(Y22<0.75,W23,W24))),IF(Y22<0.5,X21,X22))","v":"Delmi"},"selected patient",{"$f":"IF($Y$23=W23,1,0)","v":0}]
["Is that SRS? Only if the hospitals have exactly the same number of patients.",null,null,null,null,null,null,null,null,null,null,null,null,"CPS is \"sample one hospital, then sample one patient\"",null,null,null,null,null,null,null,null,"Delmi",null,null,null,{"$f":"IF($Y$23=W24,1,0)","v":1}]
["If they do not, then patients in smaller hospitals are more likely to be chosen. Look at the toy example to the right. -->",null,null,null,null,null,null,null,null,null,null,null,null,"If you are in A, you have a 1/2 times 1/4 chance of being selected:",null,null,null,null,null,null,null,null,null,null,null,null,"Track Rolando and Jose in a sim (using the button below)."]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":"(1/2)*(1/4)","v":0.125},null,null,null,null,null,null,null,null,null,null,null,"What do you get?"]
["CPS is a complex sampling scheme. There is no list of 340M people in the US.",null,null,null,null,null,null,null,null,null,null,null,null,"If you are in B, you have a 1/2 times 1/2 chance of being selected:"]
["The CPS has a super complicated process of strata (like states) and clusters (city blocks).",null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":"(1/2)*(1/2)","v":0.25}]
["You are much more likely to be selected if you are in Wyoming or you are Hispanic.",null,null,null,null,null,null,null,null,null,null,null,null,"In upcoming work, we take the inverse (reciprocal) of the chances of being selected to compute a probability weight:"]
[null,null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":"1/O26","v":8},"people represented by a sample A patient"]
["So the big question here is: \"What is the effect of unequal probability of selection on OLS?\" ",null,null,null,null,null,null,null,null,null,null,null,null,null,{"$f":"1/O28","v":4},"people represented by a sample B patient"]
["Before we find out the answer, let's make sure we know how violations of a well-behaved DGP can affect OLS:",null,null,null,null,null,null,null,null,null,null,null,null,"\"But there aren’t 8*4 + 4*2 = 40 patients!\""]
[null,null,null,null,null,null,null,null,null,null,null,null,null,"Right, you NEVER sum weights of all patients. Remember, 1/2 chance of A and B so 0.5*8 + 0.5*4 = 6."]
["What two bad things can happen when you use OLS on a DGP that does not meet the requirements for OLS to be BLUE?",null,null,null,null,null,null,null,null,null,null,null,null,"If the sampled patient happens to be from A, that person represents 8 people; if from B, that person represents 4 people."]
[null,null,null,null,null,null,null,null,null,null,null,null,null,"But this does not mean the population is 8 or 4 or 8 + 4 = 12."]
[null,null,null,null,null,null,null,null,null,null,null,null,null,"It means the one sampled unit is “inflated” to represent the population."]

{"sheet":"ChatGPT","index":3,"n_rows":76,"n_cols":7,"rows_kept":76,"cut":false,"hidden":true,"merged":[]}
["What two bad things can happen when you use OLS on a DGP that does not meet the requirements for OLS to be BLUE?"]
[]
["Citation: ChatGPT's reply on November 20, 2025:"]
[]
["Two distinct “bad things” can happen, depending on which OLS assumptions fail. They fall into two separate categories:"]
[]
["1. Loss of Unbiasedness (or Consistency) — OLS no longer targets the truth"]
[]
["This happens when the exogeneity assumption fails:",null,null,null,null,null,"A simple interpretation of the \"exogeneity assumption\": \"the included Xs cannot be connected to the error term,\" exogenous means \"outside the system\" like an exogenous variable is fixed and does not change as endoegnous variables are changed"]
[]
["Examples: omitted variables, simultaneity, measurement error in regressors, sample selection, reverse causality."]
[]
["Bad consequence:"]
[]
["OLS is biased and inconsistent."]
[]
["You may get the wrong sign, the wrong size, or completely misleading estimates."]
[]
["This is the big problem—OLS stops telling you the correct causal or structural effect."]
[]
[]
[]
["2. Loss of Efficiency (and wrong standard errors) — inference becomes invalid"]
[]
["This happens when OLS is still unbiased but not BLUE:"]
[]
["Heteroskedasticity"]
[]
["Serial correlation"]
[]
["Clustered errors"]
[]
["Non-spherical error covariance"]
[]
["Bad consequences:"]
[]
["(a) OLS is no longer efficient"]
[]
["It is still unbiased (if exogeneity holds),"]
[]
["but it is not minimum variance among linear unbiased estimators."]
[]
["You could do better with GLS or robust methods."]
[]
["(b) OLS standard errors become wrong"]
[]
["Because OLS formulas assume homoskedastic, uncorrelated errors, violation means:"]
[]
["Standard errors are biased (usually too small)."]
[]
["(t)-tests are invalid."]
[]
["Confidence intervals are wrong."]
[]
["Hypothesis tests give incorrect significance."]
[]
["This is often the practical bad outcome even when point estimates are fine."]
[]
[]
[]
["In short: the two bad things"]
[]
["1. The OLS coefficient estimates can be wrong (biased/inconsistent)."]
[]
["2. The OLS inference can be wrong (inefficient estimator, wrong standard errors → invalid hypothesis tests)."]
[]
["NEXT STEP"]
[]
["Go to the DGP sheet and refer to the detailed documentation in the Doc sheet or the published paper (link at top in Intro sheet), if needed."]
["Read and scroll right in the DGP sheet and at the end (column AF) you will see a series of tasks that will show you the effect of unequal probability of selection on OLS."]
["There are a series of simulations you can run to demonstrate the effect of unequal probability of selection on OLS -- not surprisingly, it causes trouble."]
["The specific trouble depends on the kind of unequal probability of selection."]
[]
["Use the Refresh button (instead of F9) to see changes in the chart in the DGP sheet."]
[]
["PLEASE PROCEED TO THE DGP SHEET"]

{"sheet":"DGP","index":4,"n_rows":1002,"n_cols":62,"rows_kept":1002,"cut":false,"hidden":true,"merged":[[0,0,0,1],[0,11,0,13],[0,5,0,7],[0,21,0,27]]}
["DGP Parameters",null,"Schooling (Years)","Wage ($/hr)",null,"Describing the Population",null,null,null,null,"Equal","Assigning Probability",null,null,"Unequal (Schooling)","Unequal (Wage)",null,"Prob 1st Draw","Schooling (Years)","Wage ($/hr)","Weight (WR)",{"$f":"sampletype(R2)","v":"A Single Sample: Equal Prob, WR"},null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,"schooling","true ","ols","wls","bubble size"]
["slope",2,14,24.49,null,null,null,null,null,null,{"$f":"1/1000","v":1E-3},null,"Prob ","Prob",{"$f":"IF(C2=12,$M$4,IF(C2=13,$M$5,IF(C2=14,$M$6,IF(C2=15,$M$7,$M$8))))","v":4.7156372549026095E-4},{"$f":"IF(D2<20,$M$14,IF(D2<24,$M$15,IF(D2<28,$M$16,IF(D2<32,$M$17,$M$18))))","v":7.8587786259546764E-4},{"$f":"IF(AND(S2=12,T2=7.97),1,0)","v":0},{"$f":"[1]!sample(C2:D1001,K2:K1001,100,1)","v":1E-3},15,30.23,{"$f":"1/(1-((1-R2)^100))","v":10.503335278386448},"Estimated slope via OLS",null,null,null,"Estimated slope via probW LS",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,11.5,{"$f":"$I$8+$I$10*BF2","v":22.671741979710372},{"$f":"IF($AD$9=1,$W$3+$V$3*BF2,#N/A)","v":{"$error":"#N/A"}},{"$f":"IF($AD$12=1,$AA$3+$Z$3*BF2,#N/A)","v":{"$error":"#N/A"}},10]
["SD",5,14,26.46,null,null,null,null,null,null,{"$f":"1/1000","v":1E-3},"Schooling","1st Draw","Inclusion (WR)",4.7156372549026095E-4,7.8587786259546764E-4,{"$f":"IF(AND(S3=12,T3=7.97),1,0)","v":0},1E-3,12,13.81,{"$f":"1/(1-((1-R3)^100))","v":10.503335278386448},{"$f":"LINEST(T2:T101,S2:S101,1,1)","v":2.5852020202020207},-8.008328282828284,"b1",null,{"$f":"LINESTw(T2:T101,S2:S101,U2:U101)","v":2.5852020202018839},-8.0083282828267404,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,11.51,{"$f":"$I$8+$I$10*BF3","v":22.691998469489651},{"$f":"IF($AD$9=1,$W$3+$V$3*BF3,#N/A)","v":{"$error":"#N/A"}},{"$f":"IF($AD$12=1,$AA$3+$Z$3*BF3,#N/A)","v":{"$error":"#N/A"}},10]
[null,null,13,24.09,null,null,null,null,null,null,1E-3,12,3.0000000000000001E-3,{"$f":"(1-((1-M4)^100))","v":0.25951574046021642},1E-3,7.8587786259546764E-4,0,1E-3,16,33.840000000000003,10.503335278386448,0.35549663282986249,5.0020283807319812,{"$f":"SLOPE($D$2:$D$1001,$C$2:$C$1001)","v":2.0256489779277524},null,0.36701956278088088,5.232699350541008,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,11.52,22.712254959268925,{"$error":"#N/A"},{"$error":"#N/A"},10]
["Example",null,15,26.08,null,null,null,null,null,null,1E-3,13,1E-3,{"$f":"(1-((1-M5)^100))","v":9.5207852886290301E-2},4.0000000000000002E-4,7.8587786259546764E-4,0,1E-3,16,27.28,10.503335278386448,0.35049111386574827,5.00228102756805,null,null,0.35049111386574827,5.0022810275680527,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,11.53,22.732511449048204,{"$error":"#N/A"},{"$error":"#N/A"},10]
[{"$f":"ROUND(RAND()*5+11.5,0)","v":12},{"$f":"ROUND(slope*A6+NORMINV(RAND(),0,SD),2)","v":23.16},13,31.29,null,null,null,null,null,null,1E-3,14,4.7156372549026095E-4,{"$f":"(1-((1-M6)^100))","v":4.6072393574350445E-2},1E-3,4.0000000000000002E-4,0,1E-3,14,29.42,10.503335278386448,52.88323207289217,98,"Use F9 or Refresh (in chart)",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,11.54,22.752767938827482,{"$error":"#N/A"},{"$error":"#N/A"},10]
["Press F9 to draw a new",null,15,35.49,null,null,null,null,"b0",null,1E-3,15,4.0000000000000002E-4,{"$f":"(1-((1-M7)^100))","v":3.9218249182720433E-2},4.0000000000000002E-4,1E-4,0,1E-3,14,33.950000000000003,10.503335278386448,1323.2873580808077,2452.2359169191923,"to draw another sample",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,11.55,22.77302442860676,{"$error":"#N/A"},{"$error":"#N/A"},10]
["observation above",null,15,27.42,null,null,null,null,{"$f":"INTERCEPT($D$2:$D$1001,$C$2:$C$1001)","v":-0.62322126645878129},null,1E-3,16,2.0000000000000001E-4,{"$f":"(1-((1-M8)^100))","v":1.9803287350054655E-2},4.0000000000000002E-4,7.8587786259546764E-4,0,1E-3,15,20.18,10.503335278386448,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,11.56,22.793280918386039,{"$error":"#N/A"},{"$error":"#N/A"},10]
[null,null,14,25.53,null,null,null,null,"b1",null,1E-3,null,null,null,4.7156372549026095E-4,7.8587786259546764E-4,0,1E-3,13,19.18,10.503335278386448,null,null,null,"Schooling","Pop","OLS","W LS",null,0,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,11.57,22.813537408165317,{"$error":"#N/A"},{"$error":"#N/A"},10]
["The data in columns",null,12,20.11,null,null,null,null,{"$f":"SLOPE($D$2:$D$1001,$C$2:$C$1001)","v":2.0256489779277524},null,1E-3,"pop sum prob",{"$f":"SUM(O2:O1001)","v":0.99999900000000996},null,3.0000000000000001E-3,1.5E-3,0,1E-3,12,26.84,10.503335278386448,null,null,null,12,{"$f":"INTERCEPT($D$2:$D$1001,$C$2:$C$1001)+SLOPE($D$2:$D$1001,$C$2:$C$1001)*Y10","v":23.684566468674248},{"$f":"$W$3+$V$3*Y10","v":23.014095959595966},{"$f":"$AA$3+$Z$3*Y10","v":23.014095959595871},null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,11.58,22.833793897944592,{"$error":"#N/A"},{"$error":"#N/A"},10]
["C and D were created",null,13,25.09,null,null,null,null,"Approximately 2.02565 ",null,1E-3,null,null,null,1E-3,7.8587786259546764E-4,0,1E-3,16,25.68,10.503335278386448,null,null,null,16,{"$f":"INTERCEPT($D$2:$D$1001,$C$2:$C$1001)+SLOPE($D$2:$D$1001,$C$2:$C$1001)*Y11","v":31.787162380385258},{"$f":"$W$3+$V$3*Y11","v":33.354904040404051},{"$f":"$AA$3+$Z$3*Y11","v":33.354904040403405},null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,11.59,22.85405038772387,{"$error":"#N/A"},{"$error":"#N/A"},10]
["by using the formulas in",null,15,26.74,null,null,null,null,null,null,1E-3,null,"Prob ","Prob",4.0000000000000002E-4,7.8587786259546764E-4,0,1E-3,13,30.9,10.503335278386448,null,null,null,null,null,null,null,null,0,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,11.6,22.874306877503145,{"$error":"#N/A"},{"$error":"#N/A"},10]
["cells A6 and B6, then",null,13,29.36,null,null,null,null,null,null,1E-3,"Wage Prob","1st Draw","Inclusion (WR)",1E-3,4.0000000000000002E-4,0,1E-3,12,22.9,10.503335278386448,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,11.61,22.894563367282423,{"$error":"#N/A"},{"$error":"#N/A"},10]
["copying and pasting as",null,14,29.42,null,null,null,null,null,null,1E-3,"<20",5.0000000000000001E-3,{"$f":"(1-((1-M14)^100))","v":0.3942295635092713},4.7156372549026095E-4,4.0000000000000002E-4,0,1E-3,16,32.39,10.503335278386448,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,11.62,22.914819857061701,{"$error":"#N/A"},{"$error":"#N/A"},10]
["values.",null,15,21.13,null,null,null,null,null,null,1E-3,"20 to 23.99",1.5E-3,{"$f":"(1-((1-M15)^100))","v":0.13938894470362329},4.0000000000000002E-4,1.5E-3,0,1E-3,16,31.97,10.503335278386448,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,11.63,22.93507634684098,{"$error":"#N/A"},{"$error":"#N/A"},10]
["This produces a fixed",null,12,28.93,null,null,"Data",null,null,null,1E-3,"24 to 27.99",7.8587786259546764E-4,{"$f":"(1-((1-M16)^100))","v":7.5607655910688076E-2},3.0000000000000001E-3,4.0000000000000002E-4,0,1E-3,14,26.21,10.503335278386448,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,11.64,22.955332836620258,{"$error":"#N/A"},{"$error":"#N/A"},10]
["finite population from",null,12,14.97,null,"Schooling (Years)","Count of Wage ($/hr)","Average of Wage ($/hr)","StdDevp of Wage ($/hr)",null,1E-3,"28 to 31.99",4.0000000000000002E-4,{"$f":"(1-((1-M17)^100))","v":3.9218249182720433E-2},3.0000000000000001E-3,5.0000000000000001E-3,0,1E-3,14,30.12,10.503335278386448,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,11.65,22.975589326399536,{"$error":"#N/A"},{"$error":"#N/A"},10]
["which samples will be",null,15,23.72,null,12,196,23.677959183673476,4.6858466791364659,null,1E-3,"32+",1E-4,{"$f":"(1-((1-M18)^100))","v":9.9506613086266515E-3},4.0000000000000002E-4,1.5E-3,0,1E-3,13,28.51,10.503335278386448,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,11.66,22.995845816178811,{"$error":"#N/A"},{"$error":"#N/A"},10]
["taken.",null,15,33.94,null,13,195,25.981948717948715,4.6606168071416523,null,1E-3,null,null,null,4.0000000000000002E-4,1E-4,0,1E-3,15,26.63,10.503335278386448,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,11.67,23.016102305958089,{"$error":"#N/A"},{"$error":"#N/A"},10]
[null,null,14,38.06,null,14,204,27.63196078431374,5.1508390786499838,null,1E-3,"pop sum prob",{"$f":"SUM(P2:P1001)","v":0.99999999999999722},null,4.7156372549026095E-4,1E-4,0,1E-3,14,26.13,10.503335278386448,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,11.68,23.036358795737367,{"$error":"#N/A"},{"$error":"#N/A"},10]
[null,null,12,25.14,null,15,199,29.201758793969866,5.1541903157688669,null,1E-3,null,null,null,3.0000000000000001E-3,7.8587786259546764E-4,0,1E-3,14,20.010000000000002,10.503335278386448,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,11.69,23.056615285516642,{"$error":"#N/A"},{"$error":"#N/A"},10]
[null,null,13,17.059999999999999,null,16,206,32.179854368932055,4.7856974898365037,null,1E-3,null,null,null,1E-3,5.0000000000000001E-3,0,1E-3,15,30.86,10.503335278386448,"Click Refresh (in chart) to update the table below (F9 will not recalculate this PivotTable).",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,11.7,23.07687177529592,{"$error":"#N/A"},{"$error":"#N/A"},10]
[null,null,14,27.47,null,"Grand Total",1000,27.784480000000006,5.6814844301116034,null,1E-3,null,null,null,4.7156372549026095E-4,7.8587786259546764E-4,0,1E-3,16,35.380000000000003,10.503335278386448,null,null,"Count of Schooling (Years)",null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,11.71,23.097128265075202,{"$error":"#N/A"},{"$error":"#N/A"},10]
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[null,null,12,22.18,null,null,null,null,null,null,1E-3,null,null,null,3.0000000000000001E-3,1.5E-3]
[null,null,14,25.61,null,null,null,null,null,null,1E-3,null,null,null,4.7156372549026095E-4,7.8587786259546764E-4]
[null,null,16,36.35,null,null,null,null,null,null,1E-3,null,null,null,2.0000000000000001E-4,1E-4]
[null,null,16,37.119999999999997,null,null,null,null,null,null,1E-3,null,null,null,2.0000000000000001E-4,1E-4]
[null,null,16,33.659999999999997,null,null,null,null,null,null,1E-3,null,null,null,2.0000000000000001E-4,1E-4]
[null,null,12,24.84,null,null,null,null,null,null,1E-3,null,null,null,3.0000000000000001E-3,7.8587786259546764E-4]
[null,null,12,17.059999999999999,null,null,null,null,null,null,1E-3,null,null,null,3.0000000000000001E-3,5.0000000000000001E-3]
[null,null,12,19.09,null,null,null,null,null,null,1E-3,null,null,null,3.0000000000000001E-3,5.0000000000000001E-3]
[null,null,16,35.89,null,null,null,null,null,null,1E-3,null,null,null,2.0000000000000001E-4,1E-4]
[null,null,13,29,null,null,null,null,null,null,1E-3,null,null,null,1E-3,4.0000000000000002E-4]
[null,null,13,24.08,null,null,null,null,null,null,1E-3,null,null,null,1E-3,7.8587786259546764E-4]
[null,null,15,31.77,null,null,null,null,null,null,1E-3,null,null,null,4.0000000000000002E-4,4.0000000000000002E-4]
[null,null,14,20.57,null,null,null,null,null,null,1E-3,null,null,null,4.7156372549026095E-4,1.5E-3]
[null,null,12,27.94,null,null,null,null,null,null,1E-3,null,null,null,{"$f":"IF(C962=12,$M$4,IF(C962=13,$M$5,IF(C962=14,$M$6,IF(C962=15,$M$7,$M$8))))","v":3.0000000000000001E-3},{"$f":"IF(D962<20,$M$14,IF(D962<24,$M$15,IF(D962<28,$M$16,IF(D962<32,$M$17,$M$18))))","v":7.8587786259546764E-4}]
[null,null,15,26.5,null,null,null,null,null,null,{"$f":"1/1000","v":1E-3},null,null,null,4.0000000000000002E-4,7.8587786259546764E-4]
[null,null,16,23,null,null,null,null,null,null,1E-3,null,null,null,2.0000000000000001E-4,1.5E-3]
[null,null,12,20.89,null,null,null,null,null,null,1E-3,null,null,null,3.0000000000000001E-3,1.5E-3]
[null,null,14,34.19,null,null,null,null,null,null,1E-3,null,null,null,4.7156372549026095E-4,1E-4]
[null,null,16,32.54,null,null,null,null,null,null,1E-3,null,null,null,2.0000000000000001E-4,1E-4]
[null,null,12,33.380000000000003,null,null,null,null,null,null,1E-3,null,null,null,3.0000000000000001E-3,1E-4]
[null,null,16,30.26,null,null,null,null,null,null,1E-3,null,null,null,2.0000000000000001E-4,4.0000000000000002E-4]
[null,null,12,20.18,null,null,null,null,null,null,1E-3,null,null,null,3.0000000000000001E-3,1.5E-3]
[null,null,12,27.78,null,null,null,null,null,null,1E-3,null,null,null,3.0000000000000001E-3,7.8587786259546764E-4]
[null,null,16,24.36,null,null,null,null,null,null,1E-3,null,null,null,2.0000000000000001E-4,7.8587786259546764E-4]
[null,null,14,19.54,null,null,null,null,null,null,1E-3,null,null,null,4.7156372549026095E-4,5.0000000000000001E-3]
[null,null,14,30.43,null,null,null,null,null,null,1E-3,null,null,null,4.7156372549026095E-4,4.0000000000000002E-4]
[null,null,15,31.23,null,null,null,null,null,null,1E-3,null,null,null,4.0000000000000002E-4,4.0000000000000002E-4]
[null,null,12,24.17,null,null,null,null,null,null,1E-3,null,null,null,3.0000000000000001E-3,7.8587786259546764E-4]
[null,null,15,28.93,null,null,null,null,null,null,1E-3,null,null,null,4.0000000000000002E-4,4.0000000000000002E-4]
[null,null,15,29.73,null,null,null,null,null,null,1E-3,null,null,null,4.0000000000000002E-4,4.0000000000000002E-4]
[null,null,13,22.76,null,null,null,null,null,null,1E-3,null,null,null,1E-3,1.5E-3]
[null,null,16,34.57,null,null,null,null,null,null,1E-3,null,null,null,2.0000000000000001E-4,1E-4]
[null,null,16,41.29,null,null,null,null,null,null,1E-3,null,null,null,2.0000000000000001E-4,1E-4]
[null,null,12,22.92,null,null,null,null,null,null,1E-3,null,null,null,3.0000000000000001E-3,1.5E-3]
[null,null,15,26.42,null,null,null,null,null,null,1E-3,null,null,null,4.0000000000000002E-4,7.8587786259546764E-4]
[null,null,12,23.41,null,null,null,null,null,null,1E-3,null,null,null,3.0000000000000001E-3,1.5E-3]
[null,null,13,24.11,null,null,null,null,null,null,1E-3,null,null,null,1E-3,7.8587786259546764E-4]
[null,null,12,25.28,null,null,null,null,null,null,1E-3,null,null,null,3.0000000000000001E-3,7.8587786259546764E-4]
[null,null,16,31.01,null,null,null,null,null,null,1E-3,null,null,null,2.0000000000000001E-4,4.0000000000000002E-4]
[null,null,15,31.34,null,null,null,null,null,null,1E-3,null,null,null,4.0000000000000002E-4,4.0000000000000002E-4]
[null,null,16,41.55,null,null,null,null,null,null,1E-3,null,null,null,2.0000000000000001E-4,1E-4]
[null,null,14,18.48,null,null,null,null,null,null,1E-3,null,null,null,4.7156372549026095E-4,5.0000000000000001E-3]
[null,null,13,17.829999999999998,null,null,null,null,null,null,1E-3,null,null,null,1E-3,5.0000000000000001E-3]
[null,null,12,20.61,null,null,null,null,null,null,1E-3,null,null,null,3.0000000000000001E-3,1.5E-3]
[null,null,14,27.22,null,null,null,null,null,null,1E-3,null,null,null,4.7156372549026095E-4,7.8587786259546764E-4]
[null,null,13,32.71,null,null,null,null,null,null,1E-3,null,null,null,1E-3,1E-4]
[null,null,15,32.700000000000003,null,null,null,null,null,null,1E-3,null,null,null,4.0000000000000002E-4,1E-4]
[null,null,16,27.28,null,null,null,null,null,null,1E-3,null,null,null,2.0000000000000001E-4,7.8587786259546764E-4]
[null,null,16,34.630000000000003,null,null,null,null,null,null,1E-3,null,null,null,2.0000000000000001E-4,1E-4]
[null,null,16,40.85,null,null,null,null,null,null,1E-3,null,null,null,2.0000000000000001E-4,1E-4]
[null,null,16,37.68,null,null,null,null,null,null,1E-3,null,null,null,2.0000000000000001E-4,1E-4]
[null,null,13,23.19,null,null,null,null,null,null,1E-3,null,null,null,1E-3,1.5E-3]
[null,null,13,26.33,null,null,null,null,null,null,1E-3,null,null,null,1E-3,7.8587786259546764E-4]
[null,null,null,null,null,null,null,null,null,null,{"$f":"SUM(K2:K1001)","v":1.0000000000000007},null,null,null,{"$f":"SUM(O2:O1001)","v":0.99999900000000996},{"$f":"SUM(P2:P1001)","v":0.99999999999999722}]

{"sheet":"Doc","index":5,"n_rows":33,"n_cols":15,"rows_kept":33,"cut":false,"hidden":false,"merged":[]}
[]
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[null,null,null,null,null,null,null,null,null,null,null,null,null,null,"Screenshot of the dialog box produced by clicking the MC Sim button."]

{"sheet":"SampleFn","index":6,"n_rows":51,"n_cols":20,"rows_kept":51,"cut":false,"hidden":true,"merged":[[17,12,17,13],[17,14,17,15],[34,12,34,14]]}
["This sheet explains how the SAMPLE(PopRange, ProbRange, n, Optional WithRep) function works",null,null,null,null,null,null,null,"To use the SAMPLE function:",null,null,"1) Enter population data, X(s) and Y, and probability of selection on the 1st draw (prob column need not be contiguous with pop data)"]
[null,null,null,null,null,null,null,null,null,null,null,"2) Select a n x pop col (including prob) range of cells"]
["Be sure to click on cells to see the formulas being used; press F9 to recalculate ",null,null,null,null,null,null,null,null,null,null,"3) Type the formula in the formula bar, =sample(cell range with pop data, cell range of a single column with probs, sample size value which should be the same as the number of rows selected in step 2, a 1 if sampling with replacement or a 0 if sampling without replacement)"]
[null,null,null,null,null,null,null,null,null,null,null,"4) Simultaneously hold down Ctrl-Shift and then press the Enter key --> function outputs the obs chosen in the sample, with the probability of selection on the 1st draw"]
[null,null,null,null,null,null,null,null,"Click ON cells I9, M9, and K9 to see how the sample function works"]
["Excel's RAND() function is U[0,1]",null,null,null,null,null,null,null,"Click IN a sample function cell to edit it, Ctrl-Shift-Enter to input; Esc to exit"]
[null,{"$f":"RAND()","v":0.2113526450568064},"press F9",null,"Population",null,"Prob",null,"Sample -- draw 2 from pop w rep",null,null,null,"Sample -- draw 2 from pop w/o rep",null,null,null,null,"Sample -- draw 3 from pop w/o rep"]
[null,"The values above bounce around in the [0,1]",null,null,"X","Y","1st Draw",null,"Prob","X","Y",null,"Prob","X","Y",null,null,"Prob","X","Y"]
[null,"as you press F9.",null,null,10,56,0.4,null,{"$f":"[1]!sample(E9:F12,G9:G12,2,1)","v":0.3},10.3,57,null,{"$f":"[1]!sample(E9:F12,G9:G12,2,0)","v":0.2},9.9,58,null,null,{"$f":"[1]!sample(E9:F12,G9:G12,3,0)","v":0.3},10.3,57]
[null,null,null,null,10.3,57,0.3,null,0.3,10.3,57,null,0.4,10,56,null,null,0.2,9.9,58]
[null,null,null,null,9.9,58,0.2,null,null,null,null,null,null,null,null,null,null,0.4,10,56]
[null,null,null,null,10.199999999999999,58,0.1,null,"Was the first obs chosen?",null,null,null,"Was the first obs chosen?",null,null,null,null,"Was the first obs chosen?"]
[null,null,null,null,null,null,{"$f":"SUM(G9:G12)","v":0.99999999999999989},null,{"$f":"IF(OR(I9=0.4,I10=0.4),1,0)","v":0},null,null,null,{"$f":"IF(OR(M9=0.4,M10=0.4),1,0)","v":1},null,null,null,null,{"$f":"IF(OR(R9=0.4,R10=0.4,R11=0.4),1,0)","v":1}]
[null,null,null,null,null,null,"Probs should sum to 1.",null,null,null,null,null,"Testing the SAMPLE function with and without replacament"]
[null,null,null,null,null,null,null,null,null,null,null,null,"Click the MSSim button to track cells I13 and M13"]
[null,null,null,null,null,null,null,null,null,null,null,null,"to check the SAMPLE function. See right for a picture -->"]
[null,null,null,null,null,null,null,null,null,null,null,null,"Here are our results for 10,000 samples (repetitions):"]
["SAMPLE function pseudo-code (full code available in the Visual Basic module in this workbook, press Alt-F11)",null,null,null,null,null,null,null,null,null,null,null,"SampleFn!$I$13",null,"SampleFn!$M$13"]
[null,null,null,null,null,null,null,null,null,null,null,null,"Average",0.63739999999999997,"Average",0.72140000000000004,"+/- 0.004 (=0.4483/sqrt(10,000))"]
[null,null,null,null,null,null,null,null,null,null,null,null,"SD",0.48077474396586284,"SD",0.44833262229512755]
[null,null,null,null,null,null,null,null,null,null,null,null,"Max",1,"Max",1]
[null,null,null,null,null,null,null,null,null,null,null,null,"Min",0,"Min",0]
[null,null,null,null,null,null,null,null,null,null,null,null,"The exact probability of including the first obs when sampling WITH replacement is"]
[null,null,null,null,null,null,null,null,null,null,null,null,{"$f":"1-0.6*0.6","v":0.64},"= 1 - prob 1st obs not taken in 1st draw*prob 1st obs not taken in 2nd draw"]
[null,null,null,null,null,null,null,null,null,null,null,null,"This agrees with the simulation result in cell J19 above."]
[null,null,null,null,null,null,null,null,null,null,null,null,"The exact probability of including the first obs when sampling WITHOUT replacement is"]
[null,null,null,null,null,null,null,null,null,null,null,null,0.4,"prob 1st obs taken in 1st draw"]
[null,null,null,null,null,null,null,null,null,null,null,null,{"$f":"0.3*(0.4/0.7)","v":0.17142857142857146},"prob 2nd obs taken in 1st draw and 1st obs taken in 2nd draw"]
[null,null,null,null,null,null,null,null,null,null,null,null,{"$f":"0.2*(0.4/0.8)","v":0.1},"prob 3rd obs taken in 1st draw and 1st obs taken in 2nd draw"]
[null,null,null,null,null,null,null,null,null,null,null,null,{"$f":"0.1*(0.4/0.9)","v":4.4444444444444453E-2},"prob 4th obs taken in 1st draw and 1st obs taken in 2nd draw"]
[null,null,null,null,null,null,null,null,null,null,null,null,{"$f":"SUM(M27:M30)","v":0.71587301587301599}]
[null,null,null,null,null,null,null,null,null,null,null,null,"This agrees with the simulation result in cell L19 above."]
[]
[null,null,null,null,null,null,null,null,null,null,null,null,"When sampling 3 observations without replacement (traacking cell R13) yielded"]
[null,null,null,null,null,null,null,null,null,null,null,null,"Summary Statistics"]
[null,null,null,null,null,null,null,null,null,null,null,null,"Average",0.92320000000000002,null,"+/- 0.003 (=0.2663/sqrt(10,000))"]
[null,null,null,null,null,null,null,null,null,null,null,null,"SD",0.26628715869355268]
[null,null,null,null,null,null,null,null,null,null,null,null,"Max",1]
[null,null,null,null,null,null,null,null,null,null,null,null,"Min",0]
[null,null,null,null,null,null,null,null,null,null,null,null,"Tedious calculation shows that the exact expected value is"]
[null,null,null,null,null,null,null,null,null,null,null,null,{"$f":"0.7+2/9","v":0.92222222222222217}]
[null,null,null,null,null,null,null,null,null,null,null,null,"Once again, the sample function without replacement works as advertised."]
[null,null,null,null,null,null,null,null,null,null,null,null,"The exact probability of including the first obs when sampling WITH replacement is"]
["Sampling with replacement loops over the code above for as many times as needed to produce the requested sample size.",null,null,null,null,null,null,null,null,null,null,null,{"$f":"1-0.6^3","v":0.78400000000000003},"=1 - prob 1st obs not taken in 1st draw3"]
["Sampling WITHOUT replacement requires further code, which is presented and explained below.",null,null,null,null,null,null,null,null,null,null,null,"It makes sense that the probability of inclusion is greater WITHOUT than WITH replacement"]
[null,"IF SAMPLING WITHOUT REPLACEMENT, ADJUST THE INTERVAL",null,null,null,null,null,null,null,null,null,null,"because once another observation is selected, it is removed and increases the chances of selecting"]
[null,null,null,null,null,null,null,null,null,null,null,null,"the first obs."]
[null,null,null,null,null,null,null,null,null,null,null,null,"Computing the exact probability of inclusion WITHOUT replacement quickly becomes unwieldy as n rises. "]
[null,null,null,null,null,null,null,null,null,null,null,null,"In practice, WITH replacement is assumed in computing the probability of inclusion."]
[null,null,null,null,null,null,null,null,null,null,null,null,"For more detail on this issue, please see ExploringWRvWOR.xls available at"]
[null,null,null,null,null,null,null,null,null,null,null,null,"www.depauw.edu/learn/stata"]