Chapter Statistics for Business and Economics 7 th Edition. Chapter 8. Estimation: Additional Topics. Chapter Goals

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1 Chapter Statistics for Busiess a Ecoomics 7 th Eitio Chapter 8 Estimatio: Aitioal Topics Ch. 8-1 Chapter Goals After completig this chapter, ou shoul be able to: Form cofiece itervals for the ifferece betwee two meas from epeet Form cofiece itervals for the ifferece betwee two iepeet populatio meas (staar eviatios kow or ukow) Compute cofiece iterval limits for the ifferece betwee two iepeet populatio proportios Determie the require sample size to estimate a mea or proportio withi a specifie margi of error Ch. 8- Statistics for Busiess a Ecoomics, 7/eCopright 010 Pearso Eucatio, Ic. Publishig as Pretice Hall

2 Chapter 8 8- Estimatio: Aitioal Topics Chapter Topics Cofiece Itervals Populatio Meas, Depeet Samples Populatio Meas, Iepeet Samples Populatio Proportios Sample Size Determiatio Eamples: Same group before vs. after treatmet Group 1 vs. iepeet Group Proportio 1 vs. Proportio Large Populatios Fiite Populatios Ch. 8-3 Depeet Samples Depeet Tests Meas of Relate Populatios Paire or matche Repeate measures (before/after) Use ifferece betwee paire values: i = i - i Elimiates Variatio Amog Subjects Assumptios: Both Populatios Are Normall Distribute Ch. 8-4 Statistics for Busiess a Ecoomics, 7/eCopright 010 Pearso Eucatio, Ic. Publishig as Pretice Hall

3 Chapter Mea Differece Depeet The i th paire ifferece is i, where i = i - i The poit estimate for the populatio mea paire ifferece is : i1 i The sample staar eviatio is: S i1 ( ) i 1 is the umber of matche pairs i the sample Ch. 8-5 Cofiece Iterval for Mea Differece Depeet The cofiece iterval for ifferece betwee populatio meas, μ, is t 1,α/ S μ t 1,α/ S Where = the sample size (umber of matche pairs i the paire sample) Ch. 8-6 Statistics for Busiess a Ecoomics, 7/eCopright 010 Pearso Eucatio, Ic. Publishig as Pretice Hall

4 Chapter Cofiece Iterval for Mea Differece (cotiue) Depeet The margi of error is ME t 1,α/ s t -1,/ is the value from the Stuet s t istributio with ( 1) egrees of freeom for which α P(t 1 t 1, α/) Ch. 8-7 Paire Samples Eample Depeet Si people sig up for a weight loss program. You collect the followig ata: Weight: Perso Before () After () Differece, i S = i = ( ) i 1 Ch. 8-8 Statistics for Busiess a Ecoomics, 7/eCopright 010 Pearso Eucatio, Ic. Publishig as Pretice Hall

5 Chapter Paire Samples Eample (cotiue) Depeet For a 95% cofiece level, the appropriate t value is t -1,/ = t 5,.05 =.571 The 95% cofiece iterval for the ifferece betwee meas, μ, is t 1,α/ S (.571) μ μ μ t ,α/ S (.571) 6 Sice this iterval cotais zero, we caot be 95% cofiet, give this limite ata, that the weight loss program helps people lose weight Ch. 8-9 Differece Betwee Two Meas: Iepeet Samples Populatio meas, iepeet Goal: Form a cofiece iterval for the ifferece betwee two populatio meas, μ μ Differet ata sources Urelate Iepeet Sample selecte from oe populatio has o effect o the sample selecte from the other populatio The poit estimate is the ifferece betwee the two sample meas: Ch Statistics for Busiess a Ecoomics, 7/eCopright 010 Pearso Eucatio, Ic. Publishig as Pretice Hall

6 Chapter Differece Betwee Two Meas: Iepeet Samples Populatio meas, iepeet (cotiue) a kow Cofiece iterval uses z / a ukow a assume equal a assume uequal Cofiece iterval uses a value from the Stuet s t istributio Ch a Kow Populatio meas, iepeet a kow a ukow * Assumptios: Samples are raoml a iepeetl raw both populatio istributios are ormal Populatio variaces are kow Ch. 8-1 Statistics for Busiess a Ecoomics, 7/eCopright 010 Pearso Eucatio, Ic. Publishig as Pretice Hall

7 Chapter Populatio meas, iepeet a kow a ukow a Kow * Whe a are kow a both populatios are ormal, the variace of X Y is XY a the raom variable has a staar ormal istributio X ( ) (μx μy ) Z Y (cotiue) Ch Cofiece Iterval, a Kow Populatio meas, iepeet a kow a ukow * The cofiece iterval for μ μ is: X Y X ( ) zα/ μx μy ( ) zα/ Y Ch Statistics for Busiess a Ecoomics, 7/eCopright 010 Pearso Eucatio, Ic. Publishig as Pretice Hall

8 Chapter a Ukow, Assume Equal Populatio meas, iepeet a kow a ukow a assume equal a assume uequal * Assumptios: Samples are raoml a iepeetl raw Populatios are ormall istribute Populatio variaces are ukow but assume equal Ch a Ukow, Assume Equal (cotiue) Populatio meas, iepeet a kow a ukow a assume equal a assume uequal * Formig iterval estimates: The populatio variaces are assume equal, so use the two sample staar eviatios a pool them to estimate use a t value with ( + ) egrees of freeom Ch Statistics for Busiess a Ecoomics, 7/eCopright 010 Pearso Eucatio, Ic. Publishig as Pretice Hall

9 Chapter a Ukow, Assume Equal (cotiue) Populatio meas, iepeet a kow The poole variace is a ukow a assume equal a assume uequal * s p ( 1)s ( 1)s Ch Cofiece Iterval, a Ukow, Equal a ukow a assume equal a assume uequal * The cofiece iterval for μ 1 μ is: s s ( p p ) t,α/ μx μy ( ) t,α/ s p s p Where ( 1)s ( 1)s sp Ch Statistics for Busiess a Ecoomics, 7/eCopright 010 Pearso Eucatio, Ic. Publishig as Pretice Hall

10 Chapter Poole Variace Eample You are testig two computer processors for spee. Form a cofiece iterval for the ifferece i CPU spee. You collect the followig spee ata (i Mhz): CPU CPU Number Teste Sample mea Sample st ev Assume both populatios are ormal with equal variaces, a use 95% cofiece Ch Calculatig the Poole Variace The poole variace is: S p 1S 1S ( 1) ( 1) 1 56 (17-1) ( 14 1) The t value for a 95% cofiece iterval is: t, α/ t9, Ch. 8-0 Statistics for Busiess a Ecoomics, 7/eCopright 010 Pearso Eucatio, Ic. Publishig as Pretice Hall

11 Chapter Calculatig the Cofiece Limits The 95% cofiece iterval is s s ( p p ) t,α/ μx μy ( ) t,α/ s p s p ( ) (.054) μx μy ( ) (.054) μx μy We are 95% cofiet that the mea ifferece i CPU spee is betwee a Mhz. Ch. 8-1 a Ukow, Assume Uequal Populatio meas, iepeet a kow a ukow a assume equal a assume uequal * Assumptios: Samples are raoml a iepeetl raw Populatios are ormall istribute Populatio variaces are ukow a assume uequal Ch. 8- Statistics for Busiess a Ecoomics, 7/eCopright 010 Pearso Eucatio, Ic. Publishig as Pretice Hall

12 Chapter Populatio meas, iepeet a kow a ukow a assume equal a assume uequal a Ukow, Assume Uequal * (cotiue) Formig iterval estimates: The populatio variaces are assume uequal, so a poole variace is ot appropriate use a t value with egrees of freeom, where v s s s ( ) ( ) s /( 1) /( 1) Ch. 8-3 Cofiece Iterval, a Ukow, Uequal a ukow a assume equal a assume uequal * The cofiece iterval for μ 1 μ is: s s s s ( ) t,α/ μx μy ( ) t,α/ Where s s ( ) ( ) v s s /( 1) /( 1) Ch. 8-4 Statistics for Busiess a Ecoomics, 7/eCopright 010 Pearso Eucatio, Ic. Publishig as Pretice Hall

13 Chapter Two Populatio Proportios Populatio proportios Goal: Form a cofiece iterval for the ifferece betwee two populatio proportios, P P Assumptios: Both sample sizes are large (geerall at least 40 observatios i each sample) The poit estimate for the ifferece is pˆ ˆ p Ch. 8-5 Two Populatio Proportios (cotiue) Populatio proportios The raom variable Z (pˆ pˆ pˆ (1pˆ ) (p p ) ) pˆ (1pˆ ) is approimatel ormall istribute Ch. 8-6 Statistics for Busiess a Ecoomics, 7/eCopright 010 Pearso Eucatio, Ic. Publishig as Pretice Hall

14 Chapter Cofiece Iterval for Two Populatio Proportios Populatio proportios The cofiece limits for P P are: ˆ pˆ ) Z / (p pˆ (1pˆ ) pˆ (1pˆ ) Ch. 8-7 Eample: Two Populatio Proportios Form a 90% cofiece iterval for the ifferece betwee the proportio of me a the proportio of wome who have college egrees. I a raom sample, 6 of 50 me a 8 of 40 wome ha a eare college egree Ch. 8-8 Statistics for Busiess a Ecoomics, 7/eCopright 010 Pearso Eucatio, Ic. Publishig as Pretice Hall

15 Chapter Me: Eample: Two Populatio Proportios ˆ p (cotiue) Wome: ˆ p pˆ (1 p ) p (1 p ) ˆ ˆ ˆ 0.5(0.48) (0.30) For 90% cofiece, Z / = Ch. 8-9 Eample: Two Populatio Proportios (cotiue) The cofiece limits are: (pˆ pˆ ) Z α/ pˆ (1pˆ ) pˆ (1pˆ ) (.5.70) 1.645(0.101) so the cofiece iterval is < P P < Sice this iterval oes ot cotai zero we are 90% cofiet that the two proportios are ot equal Ch Statistics for Busiess a Ecoomics, 7/eCopright 010 Pearso Eucatio, Ic. Publishig as Pretice Hall

16 Chapter Sample Size Determiatio Determiig Sample Size Large Populatios Fiite Populatios For the Mea For the Proportio For the Mea For the Proportio Ch Margi of Error The require sample size ca be fou to reach a esire margi of error (ME) with a specifie level of cofiece (1 - ) The margi of error is also calle samplig error the amout of imprecisio i the estimate of the populatio parameter the amout ae a subtracte to the poit estimate to form the cofiece iterval Ch. 8-3 Statistics for Busiess a Ecoomics, 7/eCopright 010 Pearso Eucatio, Ic. Publishig as Pretice Hall

17 Chapter Sample Size Determiatio Large Populatios For the Mea zα/ Margi of Error (samplig error) ME zα/ Ch Sample Size Determiatio Large Populatios (cotiue) For the Mea ME zα/ Now solve for to get zα/ ME Ch Statistics for Busiess a Ecoomics, 7/eCopright 010 Pearso Eucatio, Ic. Publishig as Pretice Hall

18 Chapter Sample Size Determiatio (cotiue) To etermie the require sample size for the mea, ou must kow: The esire level of cofiece (1 - ), which etermies the z / value The acceptable margi of error (samplig error), ME The populatio staar eviatio, Ch Require Sample Size Eample If = 45, what sample size is eee to estimate the mea withi ± 5 with 90% cofiece? z ME (1.645) (45) 5 α/ So the require sample size is = 0 (Alwas rou up) Ch Statistics for Busiess a Ecoomics, 7/eCopright 010 Pearso Eucatio, Ic. Publishig as Pretice Hall

19 Chapter Sample Size Determiatio: Populatio Proportio Large Populatios For the Proportio pˆ zα/ p(1 ˆ p) ˆ p(1 ˆ p) ˆ ME zα/ Margi of Error (samplig error) Ch Sample Size Determiatio: Populatio Proportio Large Populatios (cotiue) For the Proportio ME zα/ p(1 ˆ p) ˆ pˆ (1p) ˆ caot be larger tha 0.5, whe pˆ = 0.5 Substitute 0.5 for pˆ (1p) ˆ a solve for to get 0.5 z ME α/ Ch Statistics for Busiess a Ecoomics, 7/eCopright 010 Pearso Eucatio, Ic. Publishig as Pretice Hall

20 Chapter Sample Size Determiatio: Populatio Proportio (cotiue) The sample a populatio proportios, pˆ a P, are geerall ot kow (sice o sample has bee take et) P(1 P) = 0.5 geerates the largest possible margi of error (so guaratees that the resultig sample size will meet the esire level of cofiece) To etermie the require sample size for the proportio, ou must kow: The esire level of cofiece (1 - ), which etermies the critical z / value The acceptable samplig error (margi of error), ME Estimate P(1 P) = 0.5 Ch Require Sample Size Eample: Populatio Proportio How large a sample woul be ecessar to estimate the true proportio efective i a large populatio withi ±3%, with 95% cofiece? Ch Statistics for Busiess a Ecoomics, 7/eCopright 010 Pearso Eucatio, Ic. Publishig as Pretice Hall

21 Chapter Solutio: Require Sample Size Eample For 95% cofiece, use z 0.05 = 1.96 ME = 0.03 Estimate P(1 P) = z ME (0.5)(1.96) (0.03) α/ (cotiue) So use = 1068 Ch Chapter Summar Compare two epeet (paire ) Forme cofiece itervals for the paire ifferece Compare two iepeet Forme cofiece itervals for the ifferece betwee two meas, populatio variace kow, usig z Forme cofiece itervals for the iffereces betwee two meas, populatio variace ukow, usig t Forme cofiece itervals for the iffereces betwee two populatio proportios Forme cofiece itervals for the populatio variace usig the chi-square istributio Determie require sample size to meet cofiece a margi of error requiremets Ch. 8-4 Statistics for Busiess a Ecoomics, 7/eCopright 010 Pearso Eucatio, Ic. Publishig as Pretice Hall