Chapter 6 Studet Lecture Notes 6-1 Busiess Statistics: A Decisio-Makig Approach 6 th Editio Chapter 6 Itroductio to Samplig Distributios Chap 6-1 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 6-2 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 6-3
Chapter 6 Studet Lecture Notes 6-2 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 6-4 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 6-5 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 98.6 99.2 0.6 degrees Chap 6-6
Chapter 6 Studet Lecture Notes 6-3 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 6-7 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 6-8 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 6-9
Chapter 6 Studet Lecture Notes 6-4 Developig a Samplig Distributio Summary Measures for the Populatio Distributio: P() i µ N.3 18 + 20 + 22 + 24 21.2 4.1 2 0 (i µ) 18 20 22 24 2.236 N A B C D Uiform Distributio Chap 6-10 Developig a Samplig Distributio Now cosider all possible samples of size 2 1 st 2 d Observatio Obs 18 20 22 24 18 18,18 18,20 18,22 18,24 20 20,18 20,20 20,22 20,24 22 22,18 22,20 22,22 22,24 24 24,18 24,20 24,22 24,24 16 possible samples (samplig with replacemet) 16 Sample Meas 1st 2d Observatio Obs 18 20 22 24 18 18 19 20 21 20 19 20 21 22 22 20 21 22 23 24 21 22 23 24 Chap 6-11 Samplig Distributio of All Sample Meas 16 Sample Meas 1st 2d Observatio Obs 18 20 22 24 18 18 19 20 21 20 19 20 21 22 22 20 21 22 23 24 21 22 23 24 Developig a Samplig Distributio P().3 0 18 19 20 21 22 23 24 (o loger uiform) Chap 6-12.2.1 Sample Meas Distributio _
Chapter 6 Studet Lecture Notes 6-5 Developig a Samplig Distributio Summary Measures of this Samplig Distributio: 18 + 19 + 21+ + 24 L N 16 i µ ( µ ) i N 2 21 L 2 2 (18-21) + (19-21) + 16 + (24-21) 2 1.58 Chap 6-13 Comparig the Populatio with its Samplig Distributio Populatio N 4 µ 21 2.236 P().3.2 Sample Meas Distributio 2 µ 21 1.58 P().3.2.1 0 18 20 22 24 A B C D.1 0 18 19 20 21 22 23 24 _ Chap 6-14 If the Populatio is Normal (THEOREM 6-1) If a populatio is ormal with mea µ ad stadard deviatio, the samplig distributio of is also ormally distributed with µ µ ad Chap 6-15
Chapter 6 Studet Lecture Notes 6-6 z-value for Samplig Distributio of Z-value for the samplig distributio of : ( µ) z where: sample mea µ populatio mea populatio stadard deviatio sample size Chap 6-16 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 6-17 Samplig Distributio Properties µ µ Normal Populatio Distributio (i.e. is ubiased ) Normal Samplig Distributio (has the same mea) µ µ Chap 6-18
Chapter 6 Studet Lecture Notes 6-7 Samplig Distributio Properties For samplig with replacemet: As icreases, decreases Larger sample size Smaller sample size Chap 6-19 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 6-20 Cetral Limit Theorem As the sample size gets large eough the samplig distributio becomes almost ormal regardless of shape of populatio Chap 6-21
Chapter 6 Studet Lecture Notes 6-8 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 6-22 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 6-23 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 6-24
Chapter 6 Studet Lecture Notes 6-9 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 3 0.5 36 Chap 6-25 Solutio : Populatio Distributio???????????? Eample 7.8-8 µ - µ 8.2-8 P(7.8 < µ < 8.2) P < < 3 3 36 36 Samplig Distributio Sample P(-0.4 < z < 0.4) 0.3108 Stadard Normal Distributio Stadardize 7.8 8.2-0.4 0.4 µ 8 µ 8 µ z 0 Chap 6-26.1554 +.1554 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 6-27
Chapter 6 Studet Lecture Notes 6-10 Samplig Distributio of p Approimated by a ormal distributio if: p 5 (1 p) 5 Samplig Distributio P(p).3.2.1 0 0. 2.4.6 8 1 p where µ p p ad p p(1 p) (where p populatio proportio) Chap 6-28 z-value 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 6-29 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 6-30
Chapter 6 Studet Lecture Notes 6-11 Eample if p.4 ad 200, what is P(.40 p.45)? p Fid : p(1 p).4(1.4) p.03464 200 Covert to stadard ormal:.40.40.45.40 P(.40 p.45) P z.03464.03464 P(0 z 1.44) Chap 6-31 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.40.45 0 1.44 p z Chap 6-32 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 6-33