Statistics for Busiess ad Ecoomics Chapter 8 Estimatio: Additioal Topics Copright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 8-1
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-
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
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
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
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
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. 9 15 Sample mea 81.31 78.61 Sample var. 60.76 48.4 Assume both populatios are ormal with equal variaces, ad use 95% cofidece Copright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 8-7
Calculatig the Pooled Variace The pooled variace is: ( S p = 1S + ( 1S ( 1+ ( 1 = 8 60.76 +14 48.4 = 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
Calculatig the Cofidece Limits The 95% cofidece iterval is ( ± t +,α / s p + s p (81.31 78.61 ± (.074 5.79 9 + 5.79 15 3.65 < µ X µ Y < 9.05 We are 95% cofidet that the mea differece i scores is betwee -3.65 ad 9.05. Copright 010 Pearso Educatio, Ic. Publishig as Pretice Hall Ch. 8-9
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
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
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
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
Eample: Two Populatio Proportios Me: 6 = = 50 0.5 (cotiued Wome: 8 = = 40 0.70 (1 (1 0.5(0.48 50 0.70(0.30 40 + = + = 0.101 For 90% cofidece, Z α/ = 1.645 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 < -0.0135 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