Sampling Error. Chapter 6 Student Lecture Notes 61. Business Statistics: A DecisionMaking Approach, 6e. Chapter Goals


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1 Chapter 6 Studet Lecture Notes 61 Busiess Statistics: A DecisioMakig Approach 6 th Editio Chapter 6 Itroductio to Samplig Distributios Chap 61 Chapter Goals After completig this chapter, you should be able to: Defie the cocept of samplig error Determie the mea ad stadard deviatio _ for the samplig distributio of the sample mea, Determie the mea ad stadard deviatio for _ the samplig distributio of the sample proportio, p Describe the Cetral Limit Theorem ad its importace Apply samplig distributios for both ad p Chap 62 Samplig Error Sample Statistics are used to estimate Populatio Parameters Problems: e: X is a estimate of the populatio mea, µ Differet samples provide differet estimates of the populatio parameter Sample results have potetial variability, thus samplig error eits Chap 63
2 Chapter 6 Studet Lecture Notes 62 Calculatig Samplig Error Samplig Error: The differece betwee a value (a statistic) computed from a sample ad the correspodig value (a parameter) computed from a populatio Eample: (for the mea) Samplig Error  µ where: sample mea µ populatio mea Chap 64 Review Populatio mea: N µ i Sample Mea: i where: µ Populatio mea sample mea i Values i the populatio or sample N Populatio size sample size Chap 65 Eample If the populatio mea is µ 98.6 degrees ad a sample of 5 temperatures yields a sample mea of 99.2 degrees, the the samplig error is µ degrees Chap 66
3 Chapter 6 Studet Lecture Notes 63 Samplig Errors Differet samples will yield differet samplig errors The samplig error may be positive or egative ( may be greater tha or less tha µ) The epected samplig error decreases as the sample size icreases Chap 67 Samplig Distributio A samplig distributio is a distributio of the possible values of a statistic for a give size sample selected from a populatio Chap 68 Developig a Samplig Distributio Assume there is a populatio Populatio size N4 Radom variable,, is age of idividuals Values of : 18, 20, 22, 24 (years) A B C D Chap 69
4 Chapter 6 Studet Lecture Notes 64 Developig a Samplig Distributio Summary Measures for the Populatio Distributio: P() i µ N (i µ) N A B C D Uiform Distributio Chap 610 Developig a Samplig Distributio Now cosider all possible samples of size 2 1 st 2 d Observatio Obs ,18 18,20 18,22 18, ,18 20,20 20,22 20, ,18 22,20 22,22 22, ,18 24,20 24,22 24,24 16 possible samples (samplig with replacemet) 16 Sample Meas 1st 2d Observatio Obs Chap 611 Samplig Distributio of All Sample Meas 16 Sample Meas 1st 2d Observatio Obs Developig a Samplig Distributio P() (o loger uiform) Chap Sample Meas Distributio _
5 Chapter 6 Studet Lecture Notes 65 Developig a Samplig Distributio Summary Measures of this Samplig Distributio: L N 16 i µ ( µ ) i N 2 21 L 2 2 (1821) + (1921) (2421) Chap 613 Comparig the Populatio with its Samplig Distributio Populatio N 4 µ P().3.2 Sample Meas Distributio 2 µ P() A B C D _ Chap 614 If the Populatio is Normal (THEOREM 61) If a populatio is ormal with mea µ ad stadard deviatio, the samplig distributio of is also ormally distributed with µ µ ad Chap 615
6 Chapter 6 Studet Lecture Notes 66 zvalue for Samplig Distributio of Zvalue for the samplig distributio of : ( µ) z where: sample mea µ populatio mea populatio stadard deviatio sample size Chap 616 Fiite Populatio Correctio Apply the Fiite Populatio Correctio if: the sample is large relative to the populatio ( is greater tha 5% of N) ad Samplig is without replacemet ( µ) z The N N 1 Chap 617 Samplig Distributio Properties µ µ Normal Populatio Distributio (i.e. is ubiased ) Normal Samplig Distributio (has the same mea) µ µ Chap 618
7 Chapter 6 Studet Lecture Notes 67 Samplig Distributio Properties For samplig with replacemet: As icreases, decreases Larger sample size Smaller sample size Chap 619 µ If the Populatio is ot Normal 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 ad the samplig distributio will have µ µ ad Chap 620 Cetral Limit Theorem As the sample size gets large eough the samplig distributio becomes almost ormal regardless of shape of populatio Chap 621
8 Chapter 6 Studet Lecture Notes 68 If the Populatio is ot Normal Samplig distributio properties: Cetral Tedecy Variatio µ µ (Samplig with replacemet) Populatio Distributio Samplig Distributio (becomes ormal as icreases) Smaller sample size Larger sample size µ Chap 622 µ How Large is Large Eough? For most distributios, > 30 will give a samplig distributio that is early ormal For fairly symmetric distributios, > 15 For ormal populatio distributios, the samplig distributio of the mea is always ormally distributed Chap 623 Eample Suppose a populatio has mea µ 8 ad stadard deviatio 3. Suppose a radom sample of size 36 is selected. What is the probability that the sample mea is betwee 7.8 ad 8.2? Chap 624
9 Chapter 6 Studet Lecture Notes 69 Solutio: Eample Eve if the populatio is ot ormally distributed, the cetral limit theorem ca be used ( > 30) so the samplig distributio of is approimately ormal µ with mea 8 ad stadard deviatio Chap 625 Solutio : Populatio Distributio???????????? Eample µ  µ P(7.8 < µ < 8.2) P < < Samplig Distributio Sample P(0.4 < z < 0.4) Stadard Normal Distributio Stadardize µ 8 µ 8 µ z 0 Chap z Populatio Proportios, p p the proportio of populatio havig some characteristic Sample proportio ( p ) provides a estimate of p: p umber of successes i the sample sample size If two outcomes, p has a biomial distributio Chap 627
10 Chapter 6 Studet Lecture Notes 610 Samplig Distributio of p Approimated by a ormal distributio if: p 5 (1 p) 5 Samplig Distributio P(p) p where µ p p ad p p(1 p) (where p populatio proportio) Chap 628 zvalue for Proportios Stadardize p to a z value with the formula: p p z p p p p(1 p) If samplig is without replacemet ad is greater tha 5% of the populatio size, the p must use the fiite populatio correctio factor: p p(1 p) N N 1 Chap 629 Eample If the true proportio of voters who support Propositio A is p.4, what is the probability that a sample of size 200 yields a sample proportio betwee.40 ad.45? i.e.: if p.4 ad 200, what is P(.40 p.45)? Chap 630
11 Chapter 6 Studet Lecture Notes 611 Eample if p.4 ad 200, what is P(.40 p.45)? p Fid : p(1 p).4(1.4) p Covert to stadard ormal: P(.40 p.45) P z P(0 z 1.44) Chap 631 Eample if p.4 ad 200, what is P(.40 p.45)? Use stadard ormal table: P(0 z 1.44).4251 Samplig Distributio Stadardized Normal Distributio.4251 Stadardize p z Chap 632 Chapter Summary Discussed samplig error Itroduced samplig distributios Described the samplig distributio of the mea For ormal populatios Usig the Cetral Limit Theorem Described the samplig distributio of a proportio Calculated probabilities usig samplig distributios Discussed samplig from fiite populatios Chap 633
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