MBACATÓLICA. Quantitative Methods. Faculdade de Ciências Económicas e Empresariais UNIVERSIDADE CATÓLICA PORTUGUESA 9. SAMPLING DISTRIBUTIONS
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1 MBACATÓLICA Quatitative Methods Miguel Gouveia Mauel Leite Moteiro Faculdade de Ciêcias Ecoómicas e Empresariais UNIVERSIDADE CATÓLICA PORTUGUESA 9. SAMPLING DISTRIBUTIONS MBACatólica 006/07 Métodos Quatitativos 7-
2 Problem! A soft-drik vedig machie is set so the amout of drik dispesed is a radom variable with a mea of 00 milliliters ad a stadard deviatio of 15 milliliters. What is the probability that the average amout dispesed i a radom sample of 36 is at least 04 milliliters: a) if the the radom variable is ormally distributed? b) if the distributio is ukow? MBACatólica 006/07 Métodos Quatitativos 7-3 Distributio of the sample mea! The sample mea (computed from observatios draw from a populatio) is a radom variable.! Our objective is to study the distributio of that variable ad to see how it is related to the distributio of the populatio from which the sample was draw. MBACatólica 006/07 Métodos Quatitativos 7-4
3 Distributio of the sample mea! Example: samples (with replacemet) of size = from a populatio with four values: 1,, 3, 4. (µ=.5 e =1.5)! Possible samples : 16 Sample meas 1,1 1, 1,3 1, ,1,,3, ,1 3, 3,3 3, ,1 4, 4,3 4, MBACatólica 006/07 Métodos Quatitativos 7-5 Distributio of the sample mea Sample Mea Total Nº of samples Probability 1/16 /16 3/16 4/16 3/16 /16 1/16 1 MBACatólica 006/07 Métodos Quatitativos 7-6
4 Distributio of the sample mea Distributio of the populatio Distributio of the sample mea f ( x) x ( x) f x MBACatólica 006/07 Métodos Quatitativos 7-7 Distributio of the sample mea E X = x. f( x ) =.5= µ! The mea of the sample mea s distributio is the mea of the populatio.! Cocepts of mea beig used: Expected value (parameter of the mea's distributio) Radom variable Parameter (parameter of the uiverse) MBACatólica 006/07 Métodos Quatitativos 7-8
5 Distributio of the sample mea ( µ ) ( ) V X = x. f x = 0.65 V X = = 1.5/! The stadard deviatio of the sample mea is:! As the sample size () icreases, the stadard deviatio of the mea decreases.! As the stadard deviatio () decreases, the stadard deviatio of the mea also decreases. = x MBACatólica 006/07 Métodos Quatitativos 7-9 Distributio of the sample mea Populatio: N = 4 Sample mea ( = ) =.5 = 1.5 EX [ ] =.5 VX [ ] = 0.65 µ f.3. ( x) f.3. ( x).1 0 x x MBACatólica 006/07 Métodos Quatitativos 7-10
6 Distributio of the sample mea X1+ X X E X = E µ + µ µ µ = = = µ X1+ X X V X = V = = = MBACatólica 006/07 Métodos Quatitativos 7-11 Distributio of the sample mea for Normal Populatios! The liear combiatio of idepedet ormal radom variables is itself a ormal radom variable.! Applicatio: If X ~ N (µ, ) the X = X i fi ~ N µ, i= 1! X Y e X/Y do ot have a ormal distributio MBACatólica 006/07 Métodos Quatitativos 7-1
7 Problem! A soft-drik vedig machie is set so the amout of drik dispesed is a radom variable with a mea of 00 milliliters ad a stadard deviatio of 15 milliliters. What is the probability that the average amout dispesed i a radom sample of 36 is at least 04 milliliters: a) if the the radom variable is ormally distributed? b) if the distributio is ukow? MBACatólica 006/07 Métodos Quatitativos 7-13 Solutio! X: quatity of the soft-drik dispesed, with µ=00 ad =15. Sample size: =36 a) X N 15 X N ( ) if ~ 00,15 ~ 00, 36! probability that the average amout is at least 04: X µ P X 04 =P =P Z 1.6 = =5.48% [ ] ad if the distributio was ukow? MBACatólica 006/07 Métodos Quatitativos 7-14
8 Cetral Limit Theorem! The distributio of a radom variable obtaied from the sum (mea) of idepedet ad idetically distributed (i.i.d) radom variables approaches a ormal distributio as icreases.! This result is idepedet from the distributio of the populatio.! If X 1, X,..., X are radom variables i.i.d. with mea µ ad variace, the: X µ ~ N ( 0,1 ) MBACatólica 006/07 Métodos Quatitativos 7-15 MBACatólica 006/07 Métodos Quatitativos 7-16
9 MBACatólica 006/07 Métodos Quatitativos 7-17 Cetral Limit Theorem As the sample size icreases the distributio of the sample mea becomes almost Normal, idepedetly of the populatio s distributio. x MBACatólica 006/07 Métodos Quatitativos 7-18
10 Cetral Limit Theorem! What sample size () is large eough? For most populatio distributios, >30 For distributios that are fairly symmetric, >15 may suffice For distributios that are ormally distributed, the samplig distributio of the mea will always be ormally distributed, regardless of the sample size. MBACatólica 006/07 Métodos Quatitativos 7-19 Solutio! X: quatity of the soft-drik dispesed, with µ=00 ad =15. Sample size: =36 b) 15 X N sice is "large" ~! = 00, 36! probability that the average amout is at least 04: P X X µ =P " P Z 1.6 = =5.48% [ ] MBACatólica 006/07 Métodos Quatitativos 7-0
11 10. INTRODUCTION TO STATISTICAL INFERENCE MBACatólica 006/07 Métodos Quatitativos 7-1 Statistical Iferece 11. Poit Estimatio 1. Cofidece Itervals 13. Hypothesis Tests MBACatólica 006/07 Métodos Quatitativos 7-
12 Problem! BakX plas to lauch a ew fiacial product differet from all the existig oes. A sample of 5 potetial ivestors provided the followig iformatio regardig the amout they wish to ivest i the ew product (ormally distributed): Σx i =1000 ad Σ(x i x) =9600. a) Compute a poit estimate for the average amout ivested. b) Compute a 90% cofidece iterval for the average amout ivested. MBACatólica 006/07 Métodos Quatitativos 7-3 Parameters ad Statistics! Parameter: is a umerical value that characterizes the distributio or the uiverse studied.! Estimator: is a radom variable that ca take differet values depedig o the particular sample draw.! Estimate: is a umber that is obtaied from a specific sample. MBACatólica 006/07 Métodos Quatitativos 7-4
13 11. Poit Estimatio MBACatólica 006/07 Métodos Quatitativos 7-5 Estimators for the mea, variace ad proportio Populatio s parameter Estimator Estimate Mea µ X x Variace S s Stadard S s deviatio Proportio p f (f ) MBACatólica 006/07 Métodos Quatitativos 7-6
14 ! Ubiasedess Estimator s properties A estimator is ubiased it the mea of its distributio equals the parameter.! Efficiecy A ubiased estimator is the most efficiet if its variace (aroud the parameter) is miimal.! Cosistecy A estimator is cosistet if, as the sample size icreases, its mea approaches the parameter ad its variace decreases. MBACatólica 006/07 Métodos Quatitativos 7-7 Ubiasedess f () Ubiased Biased µ MBACatólica 006/07 Métodos Quatitativos 7-8
15 Efficiecy f ( ) Samplig distributio of the media Samplig distributio of the mea µ MBACatólica 006/07 Métodos Quatitativos 7-9 Cosistecy f ( ) Large sample Small sample µ MBACatólica 006/07 Métodos Quatitativos 7-30
16 Problem! BakX plas to lauch a ew fiacial product differet from all the existig oes. A sample of 5 potetial ivestors provided the followig iformatio regardig the amout they wish to ivest i the ew product (ormally distributed): Σx i =1000 ad Σ(x i x) =9600. a) Compute a poit estimate for the average amout ivested. b) Compute a 90% cofidece iterval for the average amout ivested. MBACatólica 006/07 Métodos Quatitativos 7-31 Solutio! BakX plas to lauch a ew fiacial product differet from all the existig oes. A sample of 5 potetial ivestors provided the followig iformatio regardig the amout they wish to ivest i the ew product (ormally distributed): Σx i =1000 ad Σ(x i x) =9600. a) Compute a poit estimate for the average amout ivested. =5; x= =40; s = =400. poit estimate: ˆ µ = x= =40 b) Compute a 90% cofidece iterval for the average amout ivested. MBACatólica 006/07 Métodos Quatitativos 7-3
17 1. CONFIDENCE INTERVALS MBACatólica 006/07 Métodos Quatitativos 7-33 Poit Estimatio vs. Cofidece Itervals Populatio The mea, µ, is ukow Sample Radom sample Mea x = 50 I ve got 95% cofidece that µ is located betwee 40 ad 60. MBACatólica 006/07 Métodos Quatitativos 7-34
18 Cofidece Itervals for the mea! Example for a Normal populatio (or for large samples) As: X ~ N µ, we have ~ N( 0,1 ) X µ Thus: P X µ 1.96 < < 1.96 = 0.95 / MBACatólica 006/07 Métodos Quatitativos 7-35 Cofidece Itervals for the mea which ca also be writte as: P X 1.96 < µ < X = 0.95! So, we have a 95% cofidece iterval for the mea: x 1.96 < µ < x MBACatólica 006/07 Métodos Quatitativos 7-36
19 Iterpretatio of a (1-α)% cofidece iterval! (1-α)% is the percetage of cofidece itervals, from successive samples, all with size, draw from the same populatio that iclude the true value of the parameter beig estimated. MBACatólica 006/07 Métodos Quatitativos 7-37 µ z Iterpretatio of a (1-α)% cofidece iterval α / Cofidece itervals for 10 differet samples α / 1 α EX [ ] = µ α / x µ + z ( ) α / 1 α % of the itervals cotai µ ad α% do t. MBACatólica 006/07 Métodos Quatitativos 7-38
20 (1- α)% CI for the mea: Normal Pop., large ad kow! For a Normal populatio (or large ) with kow: 1. Defie the level of cofidece (1- α)%. Collect a sample with size. Compute x 3. Obtai z α/ from the statistic tables 4. The cofidece iterval is give by: x z < µ < x + z α α MBACatólica 006/07 Métodos Quatitativos 7-39 Problem! BakX plas to lauch a ew fiacial product differet from all the existig oes. A sample of 5 potetial ivestors, collected the followig iformatio regardig the amout they wish to ivest i the ew product (ormally distributed): Σx i =1000 ad Σ(x i x) =9600. a) Compute a poit estimate for the average amout ivested. b) Compute a 90% cofidece iterval for the average amout ivested. MBACatólica 006/07 Métodos Quatitativos 7-40
21 Solutio! BakX plas to lauch a ew fiacial product differet from all the existig oes. A sample of 5 potetial ivestors, collected the followig iformatio regardig the amout they wish to ivest i the ew product (ormally distributed): Σx i =1000 ad Σ(x i x) =9600. =5; x= =40; s = =400. b) Compute a 90% CI for the average amout ivested. α = 10% z0.05 = x zα < µ < x + zα 40 ± IC for µ : (33.156,46.844) MBACatólica 006/07 Métodos Quatitativos 7-41 Coflict betwee credibility ad precisio! Credibility Cofidece level of a iterval! Precisio Width of the cofidece iterval! For a give sample size : More precisio meas decrease the width of the iterval. Therefore implyig a lower level of cofidece. A higher level of cofidece implies a larger iterval (less precisio).! The oly way to icrease simultaeously the precisio ad the credibility of the iferece is to icrease. MBACatólica 006/07 Métodos Quatitativos 7-4
22 Problem! A vedig machie is calibrated to pour a quatity of liquid that follows a ormal distributio with variace equal to 16 ml. I a sample of 5 driks, the average was: x = 50ml We wat: a) To costruct a 95% Cofidece Iterval for the true average quatity of liquid o the served driks; b) To determie how may driks should be icluded o a ew sample, if the iterval precisio is to be icreased to ml. MBACatólica 006/07 Métodos Quatitativos 7-43 Solutio a) x 1.96 < µ < x x 1.96 µ < x + 1, < µ < < µ < < µ < < µ < The width of the iterval is ml. MBACatólica 006/07 Métodos Quatitativos 7-44
23 Solutio b) Width = zα = z 1.96 α = 7.84 = 6 4 MBACatólica 006/07 Métodos Quatitativos 7-45 Problem! Te aalysts have give the followig year earigs forecasts for a stock, which are ormally distributed: Forecast ( Xi) Number of aalysts ( i) Compute a 95% cofidece iterval for the populatio mea of the forecasts. MBACatólica 006/07 Métodos Quatitativos 7-46
24 Populatio s Variace ukow! Util ow we have assumed that the variace of the populatio was kow. However, it usually is ukow ad has to be estimated.! We kow that ( X ) i X i = 1 S = 1 is a ubiased estimator for the populatio variace. E S = MBACatólica 006/07 Métodos Quatitativos 7-47 Distributio of the sample mea from a Normal populatio with ukow! If the populatio is Normal, is the sample mea distributio still give by X S µ ~ N 0,1 ( )? For small samples the aswer is NO! MBACatólica 006/07 Métodos Quatitativos 7-48
25 Distributio of the sample mea from a Normal populatio with ukow! With ukow, we have a t distributio: X S µ ~ t 1 ( ) where: S = i= 1 ( x x) i 1 MBACatólica 006/07 Métodos Quatitativos 7-49 t distributio (Studet s distributio) Normal (0,1) Also bell shaped Also symmetric But with wider tails t (df = 13) t (df = 5) 0 z, t MBACatólica 006/07 Métodos Quatitativos 7-50
26 Studet s t distributio F(x) t (df = 3) if MBACatólica 006/07 Métodos Quatitativos 7-51 (1- α)% CI for the mea: Normal Pop. ad ukow! For a Normal populatio with ukow: 1. Defie the level of cofidece (1- α)%. Collect a sample with size. Compute x ( 1) 3. Obtai t α / from the statistical tables 4. The cofidece iterval is give by: s x t < µ < x+ t ( 1) ( 1) α/ α/ s MBACatólica 006/07 Métodos Quatitativos 7-5
27 Problem! Te aalysts have give the followig year earigs forecasts for a stock, which are ormally distributed: Forecast ( Xi) Number of aalysts ( i) Compute a 95% cofidece iterval for the populatio mea of the forecasts. MBACatólica 006/07 Métodos Quatitativos Solutio x = 1.45; s= ; = 10; df = 9 t = µ µ 1.47! For a 99% cofidece level, the iterval would be: µ µ MBACatólica 006/07 Métodos Quatitativos 7-54
28 Distributio of the sample mea Kow Ukow <30 30 <30 30 Normal Populatio Not Normal Populatio X µ ~ N(0,1) We do t kow the distributio X µ ~ N(0,1) X µ ~ N(0,1) CLT X µ ~ t ( 1) S We do t kow the distributio X µ ~ N(0,1) S CLT X µ ~ N(0,1) S CLT MBACatólica 006/07 Métodos Quatitativos 7-55! The true proportio of a populatio is p. The estimator of p is the proportio o the sample, i.e., Cofidece iterval for a proportio f X =, where X is a biomial variable: p E [ P] = E[ X ] = = p 1 1 p p E[ f ] = E[ X ] = = p [ ] V[ X] V f ( 1 ) ( 1 ) 1 p p p p = = = MBACatólica 006/07 Métodos Quatitativos 7-56
29 Cofidece iterval for a proportio! For a large : f p p ( 1 p) ( ) ~ N 0,1! The cofidece iterval is give by: ( 1 ) ( 1 ) f f f f f z < p< f + z α α MBACatólica 006/07 Métodos Quatitativos 7-57 (1- α)% CI for a proportio : with large samples 1. Defie the level of cofidece (1- α)%. Collect a sample of size. Compute f 3. Obtai z α/ from the statistic tables 4. The cofidece iterval is give by: ( 1 ) ( 1 ) f f f f f z < p< f + z α α MBACatólica 006/07 Métodos Quatitativos 7-58
30 Problem! We wat to estimate the proportio of voters i a political party. 400 citizes were iterviewed ad 140 of them revealed the itetio to vote o that party. Compute a 99% cofidece iterval for the proportio of votes o that party. MBACatólica 006/07 Métodos Quatitativos 7-59 Solutio = 400 f = 140 / 400 = 0.35, 1 f = α = 0.99, α / = 0.005, z =.57 α / 0.35* * p p MBACatólica 006/07 Métodos Quatitativos 7-60
31 Selectio of the sample size! The sample size is a decisio variable reflectig a coflict betwee precisio ad the cost of samplig. Very large: Too expesive Very small: Imprecise results MBACatólica 006/07 Métodos Quatitativos 7-61 Selectio of the sample size! Questio: for a desirable miimum precisio, what should be the miimum sample size to be draw? The choice of is affected by 3 factors: 1. The level of precisio or the level of margi of error (iterval width). Level of cofidece 3. The dispersio of the populatio MBACatólica 006/07 Métodos Quatitativos 7-6
32 Sample size: Estimatio of a proportio! Sice the cofidece iterval is give by: it ca also be writte as ( 1 ) ( 1 ) f f f f f z < p < f + z α α f e< p< f + e with e beig the margi of error. MBACatólica 006/07 Métodos Quatitativos 7-63 Sample size: Estimatio of a proportio! Fixig e, it is possible to obtai as: = ( z ) f α ( 1 f )! BUT: the value of f is ukow before the sample is draw. The value used for f should be the oe that maximizes p(1-p), i.e., f = 0.5. e MBACatólica 006/07 Métodos Quatitativos 7-64
33 Problem! Determie the miimum size of a sample i order to compute a 95% cofidece iterval for the proportio of cosumers who are willig to buy a ew product, with a margi of error of oe percetage poit.! Recompute that cofidece iterval if you were sure that, give the high price of the product, o more tha 5% of cosumers would buy it. MBACatólica 006/07 Métodos Quatitativos 7-65 e = 0.01 α = 5% Z α / = 1.96 Solutio = 1.96 = ! If we kew a priori that p<0.5, the = = MBACatólica 006/07 Métodos Quatitativos 7-66
34 Sample size: Estimatio of the mea! The cofidece iterval is give by: x z < µ < x + z α α ad it ca be writte as: x e< µ < x+ e Thus: = α ( z ) e MBACatólica 006/07 Métodos Quatitativos 7-67 Sample size: Estimatio of the mea! If is ukow: 1. Collect a pilot sample, with a smaller size, to estimate.. If the populatio is approximately ormal: Prob[µ ± ]=0.95 ad Prob[µ ± 3 ]=0.997 Therefore (ad usig past data or subjective evaluatios of the populatio), we ca estimate : ι. = (Percetile Percetile.5)/4 ιι. = (MAX- MIN)/6 MBACatólica 006/07 Métodos Quatitativos 7-68
35 Problem! Suppose you wat to estimate the populatio mea of the aalysts forecasts for ext year stock earigs to withi ± 0.01 with 95% cofidece. O the basis of past studies, you believe the stadard deviatio of those forecasts to be Fid the miimum sample size eeded. MBACatólica 006/07 Métodos Quatitativos 7-69 e = 0.01 = 0.03 α = 5% z α / = 1.96 Solutio 0.03 = 1.96 = We eed at least 35 forecasts i our sample. MBACatólica 006/07 Métodos Quatitativos 7-70
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