Statistics for Business and Economics

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1 Statistics for Busiess ad Ecoomics Chapter 8 Estimatio: Additioal Topics Copright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 8-1

2 8. Differece Betwee Two Meas: Idepedet Samples Populatio meas, idepedet samples Goal: Form a cofidece iterval for the differece betwee two populatio meas, µ µ Differet data sources Urelated Idepedet Sample selected from oe populatio has o effect o the sample selected from the other populatio The poit estimate is the differece betwee the two sample meas: Copright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 8-

3 σ ad σ Ukow, Assumed Equal Populatio meas, idepedet samples σ ad σ kow σ ad σ ukow σ ad σ assumed equal σ ad σ assumed uequal * Assumptios: Samples are radoml ad idepedetl draw Populatios are ormall distributed Populatio variaces are ukow but assumed equal Copright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 8-3

4 σ ad σ Ukow, Assumed Equal (cotiued Populatio meas, idepedet samples σ ad σ kow σ ad σ ukow σ ad σ assumed equal σ ad σ assumed uequal * Formig iterval estimates: The populatio variaces are assumed equal, so use the two sample stadard deviatios ad pool them to estimate σ use a t value with ( + degrees of freedom Copright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 8-4

5 σ ad σ Ukow, Assumed Equal (cotiued Populatio meas, idepedet samples σ ad σ kow The pooled variace is σ ad σ ukow σ ad σ assumed equal * s p = ( 1s + + ( 1s σ ad σ assumed uequal Copright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 8-5

6 Cofidece Iterval, σ ad σ Ukow, Equal σ ad σ ukow σ ad σ assumed equal σ ad σ assumed uequal * The cofidece iterval for µ 1 µ is: ( s s t p p p,α/ µ X µ Y ( t + + < < + +,α/ + s s p Where s p = ( 1s + ( + 1s Copright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 8-6

7 Pooled Variace Eample Stadardized tests are take b studets from large ( ad small ( high schools. Form a cofidece iterval for the differece i scores. You collect the followig data: Score Score Number Obs Sample mea Sample var Assume both populatios are ormal with equal variaces, ad use 95% cofidece Copright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 8-7

8 Calculatig the Pooled Variace The pooled variace is: ( S p = 1S + ( 1S ( 1+ ( 1 = = 5.79 The t value for a 95% cofidece iterval is: t +, α / = t, 0.05 =.074 Copright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 8-8

9 Calculatig the Cofidece Limits The 95% cofidece iterval is ( ± t +,α / s p + s p ( ± ( < µ X µ Y < 9.05 We are 95% cofidet that the mea differece i scores is betwee ad Copright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 8-9

10 8.3 Two Populatio Proportios Populatio proportios Goal: Form a cofidece iterval for the differece betwee two populatio proportios, p p Assumptios: Both sample sizes are large The poit estimate for the differece is Copright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 8-10

11 Two Populatio Proportios (cotiued Populatio proportios The radom variable Z = ( (1 (p + p (1 is approimatel ormall distributed Copright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 8-11

12 Cofidece Iterval for Two Populatio Proportios Populatio proportios The cofidece limits for p p are: ˆ ± Zα / (p (1 + (1 Copright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 8-1

13 Eample: Two Populatio Proportios Form a 90% cofidece iterval for the differece betwee the proportio of me ad the proportio of wome who have college degrees. I a radom sample, 6 of 50 me ad 8 of 40 wome had a eared college degree Copright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 8-13

14 Eample: Two Populatio Proportios Me: 6 = = (cotiued Wome: 8 = = (1 (1 0.5( ( = + = For 90% cofidece, Z α/ = Copright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 8-14

Eample: Two Populatio Proportios (cotiued The cofidece limits are: ( ± Z α/ (1 + (1 = (.5.70 ± 1.645 (0.101 so the cofidece iterval is -0.3465 < P P < -0.

15 Eample: Two Populatio Proportios (cotiued The cofidece limits are: ( ± Z α/ (1 + (1 = (.5.70 ± (0.101 so the cofidece iterval is < P P < Sice this iterval does ot cotai zero we are 90% cofidet that the two proportios are ot equal Copright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 8-15

Chapter 8. Confidence Interval Estimation. Copyright 2015, 2012, 2009 Pearson Education, Inc. Chapter 8, Slide 1

Chapter 8. Confidence Interval Estimation. Copyright 2015, 2012, 2009 Pearson Education, Inc. Chapter 8, Slide 1 Chapter 8 Cofidece Iterval Estimatio Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 1 Learig Objectives I this chapter, you lear: To costruct ad iterpret cofidece iterval estimates for