Sampling, Sampling Distribution and Normality

Transcription

1 4/17/11 Tools of Busiess Statistics Samplig, Samplig Distributio ad ormality Preseted by: Mahedra Adhi ugroho, M.Sc Descriptive statistics Collectig, presetig, ad describig data Iferetial statistics Drawig coclusios ad/or makig decisios cocerig a populatio based oly o sample data Sources: Aderso, Sweeey,wiliams, Statistics for Busiess ad Ecoomics, 6 e, Pearso educatio ic, 7 Sugiyoo, Statistika utuk peelitia, alfbeta, Badug, 7 Statistics for Busiess ad Ecoomics, 6e 7 Pearso Educatio, Ic. Chap 7- Populatios ad s Populatio vs. A Populatio is the set of all items or idividuals of iterest Eamples: All likely voters i the et electio All parts produced today All sales receipts for ovember A is a subset of the populatio Eamples: 1 voters selected at radom for iterview A few parts selected for destructive testig Radom receipts selected for audit Populatio a b c d ef gh i jk l m o p q rs t u v w y z b c g i o r u y Statistics for Busiess ad Ecoomics, 6e 7 Pearso Educatio, Ic. Chap 7-3 Statistics for Busiess ad Ecoomics, 6e 7 Pearso Educatio, Ic. Chap 7-4 Why? Samplig Methods Less time cosumig tha a cesus Samplig Methods Less costly to admiister tha a cesus It is possible to obtai statistical results of a sufficietly high precisio based o samples. Probability samplig 1. Simple radom samplig. Proportioate stratified radom samplig 3. Disproportioate stratified radom samplig 4. Cluster samplig o probability samplig 1. Systematic samplig. Quota samplig 3. Icidetal samplig 4. Purposive samplig 5. Surfeited samplig 6. Sowball samplig Statistics for Busiess ad Ecoomics, 6e 7 Pearso Educatio, Ic. Chap 7-5 1

2 4/17/11 Makig statemets about a populatio by eamiig sample results statistics Iferetial Statistics Populatio parameters (kow) Iferece (ukow, but ca be estimated from sample evidece) Populatio Estimatio Iferetial Statistics Drawig coclusios ad/or makig decisios cocerig a populatio based o sample results. e.g., Estimate the populatio mea weight usig the sample mea weight Hypothesis Testig e.g., Use sample evidece to test the claim that the populatio mea weight is 1 pouds Statistics for Busiess ad Ecoomics, 6e 7 Pearso Educatio, Ic. Chap 7-7 Statistics for Busiess ad Ecoomics, 6e 7 Pearso Educatio, Ic. Chap 7-8 Samplig Distributios Samplig Distributios of Meas A samplig distributio is a distributio of all of the possible values of a statistic for a give size sample selected from a populatio Samplig Distributio of Mea Samplig Distributios Samplig Distributio of Proportio Samplig Distributio of Variace ote: this chapter oly discussig sample mea distributio. Statistics for Busiess ad Ecoomics, 6e 7 Pearso Educatio, Ic. Chap 7-9 Statistics for Busiess ad Ecoomics, 6e 7 Pearso Educatio, Ic. Chap 7-1 Developig a Samplig Distributio Developig a Samplig Distributio (cotiued) Assume there is a populatio Populatio size 4 Radom variable,, is age of idividuals Values of : 18,,, 4 (years) A B C D Summary Measures for the Populatio Distributio: i (i ).36 P() A B C D Uiform Distributio Statistics for Busiess ad Ecoomics, 6e 7 Pearso Educatio, Ic. Chap 7-11 Statistics for Busiess ad Ecoomics, 6e 7 Pearso Educatio, Ic. Chap 7-1

3 4/17/11 Developig a Samplig Distributio ow cosider all possible samples of size 1 st d Observatio Obs ,18 18, 18, 18,4,18,,,4,18,,,4 4 4,18 4, 4, 4,4 16 possible samples (samplig with replacemet) 16 Meas (cotiued) 1st d Observatio Obs Statistics for Busiess ad Ecoomics, 6e 7 Pearso Educatio, Ic. Chap 7-13 Developig a Samplig Distributio Samplig Distributio of All Meas 16 Meas 1st d Observatio Obs P() Meas Distributio (cotiued) Statistics for Busiess ad Ecoomics, 6e 7 Pearso Educatio, Ic. Chap (o loger uiform) Developig a Samplig Distributio Summary Measures of this Samplig Distributio: i E( ) L 1 16 (i ) (18-1) + (19-1) + L+ (4-1) 16 (cotiued) 1.58 Comparig the Populatio with its Samplig Distributio Populatio P() A B C D Meas Distributio P() Statistics for Busiess ad Ecoomics, 6e 7 Pearso Educatio, Ic. Chap 7-15 Statistics for Busiess ad Ecoomics, 6e 7 Pearso Educatio, Ic. Chap 7-16 Epected Value of Mea Stadard Error of the Mea Let 1,,... represet a radom sample from a populatio The sample mea value of these observatios is defied as 1 i i 1 Differet samples of the same size from the same populatio will yield differet sample meas A measure of the variability i the mea from sample to sample is give by the Stadard Error of the Mea: ote that the stadard error of the mea decreases as the sample size icreases Statistics for Busiess ad Ecoomics, 6e 7 Pearso Educatio, Ic. Chap 7-17 Statistics for Busiess ad Ecoomics, 6e 7 Pearso Educatio, Ic. Chap

4 4/17/11 If the Populatio is ormal If a populatio is ormal with mea ad stadard deviatio, the samplig distributio of is also ormally distributed with ad Z-value for Samplig Distributio of the Mea Z-value for the samplig distributio of : where: ( ) ( ) Z sample mea populatio mea populatio stadard deviatio sample size Statistics for Busiess ad Ecoomics, 6e 7 Pearso Educatio, Ic. Chap 7-19 Statistics for Busiess ad Ecoomics, 6e 7 Pearso Educatio, Ic. Chap 7- Fiite Populatio Correctio Samplig Distributio Properties Apply the Fiite Populatio Correctio if: a populatio member caot be icluded more tha oce i a sample (samplig is without replacemet), ad the sample is large relative to the populatio ( is greater tha about 5 of ) The Var( ) 1 or 1 (i.e. is ubiased ) ormal Populatio Distributio ormal Samplig Distributio (has the same mea) Statistics for Busiess ad Ecoomics, 6e 7 Pearso Educatio, Ic. Chap 7-1 Statistics for Busiess ad Ecoomics, 6e 7 Pearso Educatio, Ic. Chap 7- Samplig Distributio Properties For samplig with replacemet: As icreases, decreases Smaller sample size Larger sample size (cotiued) Statistics for Busiess ad Ecoomics, 6e 7 Pearso Educatio, Ic. Chap 7-3 If the Populatio is ot ormal We ca apply the Cetral Limit Theorem: Eve if the populatio is ot ormal, sample meas from the populatio will be approimately ormal as log as the sample size is large eough. Properties of the samplig distributio: ad Statistics for Busiess ad Ecoomics, 6e 7 Pearso Educatio, Ic. Chap 7-4 4

5 4/17/11 Cetral Limit Theorem If the Populatio is ot ormal (cotiued) As the sample size gets large eough the samplig distributio becomes almost ormal regardless of shape of populatio Samplig distributio properties: Cetral Tedecy Variatio Populatio Distributio Samplig Distributio (becomes ormal as icreases) Smaller sample size Larger sample size Statistics for Busiess ad Ecoomics, 6e 7 Pearso Educatio, Ic. Chap 7-5 Statistics for Busiess ad Ecoomics, 6e 7 Pearso Educatio, Ic. Chap 7-6 How Large is Large Eough? For most distributios, > 5 will give a samplig distributio that is early ormal For ormal populatio distributios, the samplig distributio of the mea is always ormally distributed Decidig Size Isaac ad Michael Approach.. P. Q s λ d ( 1) + λ. P. Q Use table o sugiyoo page 71 omogram Herry Kig Maimum sample size is Statistics for Busiess ad Ecoomics, 6e 7 Pearso Educatio, Ic. Chap 7-7 Suggested Size Proper sample size i a research are 3-5 samples Proper size i categorized sample are miimum 3 samples each category Proper sample size for multivariate data aalysis (correlatio or multivariate regressio) are miimum 1 times of variables umbers (idepedet ad depedet) Proper sample size for simple eperimetal desig that use eperimet ad cotrol groups are 1 samples each variable group. 9 ormality: ormal Curve A data set could have ormal distributio if sum of data ad stadard deviatio of upper mea ad uder mea data are same. umber Value 5

6 4/17/11 ormality: ormal Curve Divide ormal curve i 6 areas base o deviatio stadard values..7 3s s s s ( ) z i s s.7 3s ormality Test Usig Chi square Decide iterval class. I this chase use 6 as class iterval because chi square ormal distributios is divided i 6 part. Decide iterval wide Coutig the estimated chi square value ad compare the value with value that is stated i chi square table. χ ( f f ) o f e e Let me wi! If I caot be a wier, Let me brave i attempt! 6