Chater 0 Statistical Iferece About Meas ad Proortios With Two Poulatios Slide Learig objectives. Uderstad ifereces About the Differece Betwee Two Poulatio Meas: σ ad σ Kow. Uderstad Ifereces About the Differece Betwee Two Poulatio Meas: σ ad σ Ukow 3. Uderstad Ifereces About the Differece Betwee Two Poulatio Meas: Matched Samles 4. Uderstad Ifereces About the Differece Betwee Two Poulatio Proortios Slide
Overview Tests for two oulatio Two meas (Poit estimator = x x) Two roortios (Poit estimator = ) σ ad σ are kow Stadard error will be differet for each case. σ ad σ are ukow Matched (or aired) samle Slide 3 Comariso betwee oe oulatio case ad two oulatio case. Procedures of gettig cofidece iterval ad of testig for two oulatio case is very similar as those of oe oulatio case Cofidece iterval: Poit estimator ± Sigificace coefficiet * Stadard Error Test Statistics Poit estimator - Hyothesized value of arameter Stadard error Poit estimators ad stadard errors of two oulatio case are differet from those of oe oulatio case. Slide 4
Estimatig the Differece Betwee Two Poulatio Meas Let µ equal the mea of oulatio ad µ equal the mea of oulatio. The differece betwee the two oulatio meas is µ - µ. Let x equal the mea of samle ad x equal the mea of samle. The oit estimator of the differece betwee the meas of the oulatios ad is x x. x Slide 5 Samlig Distributio of x x : σ kow Exected Value E ( x x ) = µ µ Stadard Deviatio (Stadard Error) σ σ σ x = x where: σ = stadard deviatio of oulatio σ = stadard deviatio of oulatio = samle size from oulatio = samle size from oulatio Slide 6 3
Iterval Estimate Iterval Estimatio of µ - µ : σ ad σ Kow σ σ x x ± z α / where: - α is the cofidece coefficiet Slide 7 Examle: Par, Ic. Iterval Estimatio of µ - µ : σ ad σ Kow Par, Ic. is a maufacturer of golf equimet ad has develoed a ew golf ball that has bee desiged to rovide extra distace. I a test of drivig distace usig a mechaical drivig device, a samle of Par golf balls was comared with a samle of golf balls made by Ra, Ltd., a cometitor. The samle statistics aear o the ext slide. Slide 8 4
Iterval Estimatio of µ - µ : σ ad σ Kow Examle: Par, Ic. Samle Size Samle Mea Samle # Par, Ic. Samle # Ra, Ltd. 0 balls 80 balls 75 yards 58 yards Based o data from revious drivig distace tests, the two oulatio stadard deviatios are kow with σ = 5 yards ad σ = 0 yards. Slide 9 Iterval Estimatio of µ - µ : σ ad σ Kow Let us develo a 95% cofidece iterval estimate of the differece betwee the mea drivig distaces of the two brads of golf ball. Poit estimate of µ µ = x x where: = 75 58 = 7 yards µ = mea distace for the oulatio of Par, Ic. golf balls µ = mea distace for the oulatio of Ra, Ltd. golf balls Slide 0 5
Iterval Estimatio of µ - µ : σ ad σ Kow σ σ ( 5 ) ( 0 ) x x ± z α / = 7 ± 96. 0 80 7 5.4 or.86 yards to.4 yards We are 95% cofidet that the differece betwee the mea drivig distaces of Par, Ic. balls ad Ra, Ltd. balls is.86 to.4 yards. Slide Hyotheses Hyothesis Tests About µ µ : σ ad σ Kow H : µ µ D H µ µ < D 0 0 : a 0 H : µ µ D H µ µ > D 0 0 : a 0 H : H µ µ = D µ µ D 0 0 : a 0 Left-tailed Right-tailed Two-tailed Test Statistic z = ( x x ) D 0 σ σ Slide 6
Hyothesis Tests About µ µ : σ ad σ Kow Examle: Par, Ic. Ca we coclude, usig α =.0, that the mea drivig distace of Par, Ic. golf balls is greater tha the mea drivig distace of Ra, Ltd. golf balls? Slide 3 Hyothesis Tests About µ µ : σ ad σ Kow Value ad Critical Value Aroaches. Develo the hyotheses. H 0 : µ - µ < 0 H a : µ - µ > 0 where: µ = mea distace for the oulatio of Par, Ic. golf balls µ = mea distace for the oulatio of Ra, Ltd. golf balls. Secify the level of sigificace. α =.0 Slide 4 7
Hyothesis Tests About µ µ : σ ad σ Kow Value ad Critical Value Aroaches 3. Comute the value of the test statistic. ( x x ) D z = σ σ 0 (35 8) 0 7 z = = = (5) (0).6 0 80 6.49 Slide 5 Hyothesis Tests About µ µ : σ ad σ Kow Value Aroach 4. Comute the value. For z = 6.49, the value <.000. 5. Determie whether to reject H 0. Because value < α =.0, we reject H 0. At the.0 level of sigificace, the samle evidece idicates the mea drivig distace of Par, Ic. golf balls is greater tha the mea drivig distace of Ra, Ltd. golf balls. Slide 6 8
Hyothesis Tests About µ µ : σ ad σ Kow Critical Value Aroach 4. Determie the critical value ad rejectio rule. For α =.0, z.0 =.33 Reject H 0 if z >.33 5. Determie whether to reject H 0. Because z = 6.49 >.33, we reject H 0. The samle evidece idicates the mea drivig distace of Par, Ic. golf balls is greater tha the mea drivig distace of Ra, Ltd. golf balls. Slide 7 I-class exercise # (.400) (cofidece iterval) #3 (.400) (hyothesis testig) Slide 8 9
Iterval Estimatio of µ - µ : σ ad σ Ukow Whe σ ad σ are ukow, we will: use the samle stadard deviatios s ad s as estimates of σ ad σ, ad relace z α/ with t α/. Slide 9 Iterval Estimate Iterval Estimatio of µ - µ : σ ad σ Ukow s s x x ± t α / Where the degrees of freedom for t α/ are: df s s = s s Slide 0 0
Differece Betwee Two Poulatio Meas: σ ad σ Ukow Examle: Secific Motors Secific Motors of Detroit has develoed a ew automobile kow as the M car. 4 M cars ad 8 J cars (from Jaa) were road tested to comare miles-er-gallo (mg) erformace. The samle statistics are show o the ext slide. Slide Differece Betwee Two Poulatio Meas: σ ad σ Ukow Examle: Secific Motors Samle # M Cars Samle # J Cars 4 cars 8 cars 9.8 mg 7.3 mg.56 mg.8 mg Samle Size Samle Mea Samle Std. Dev. Slide
Differece Betwee Two Poulatio Meas: σ ad σ Ukow Examle: Secific Motors Let us develo a 90% cofidece iterval estimate of the differece betwee the mg erformaces of the two models of automobile. Slide 3 Poit Estimate of µ µ Poit estimate of µ µ = x x = 9.8-7.3 =.5 mg where: µ = mea miles-er-gallo for the oulatio of M cars µ = mea miles-er-gallo for the oulatio of J cars Slide 4
Iterval Estimatio of µ µ : σ ad σ Ukow The degrees of freedom for t α/ are: df (.56) (.8) 4 8 = = 4.07 = 4 (.56) (.8) 4 4 8 8 With α/ =.05 ad df = 4, t α/ =.7 Slide 5 Iterval Estimatio of µ µ : σ ad σ Ukow s s (.56) (.8) x x ± t α / = 9.8 7.3 ±.7 4 8.5.069 or.43 to 3.569 mg We are 90% cofidet that the differece betwee the miles-er-gallo erformaces of M cars ad J cars is.43 to 3.569 mg. Slide 6 3
Hyotheses Hyothesis Tests About µ µ : σ ad σ Ukow H : µ µ D H µ µ < D 0 0 : a 0 H : µ µ D H µ µ > D 0 0 : a 0 H : H µ µ = D µ µ D 0 0 : a 0 Left-tailed Right-tailed Two-tailed Test Statistic t = ( x x ) D 0 s s Slide 7 Hyothesis Tests About µ µ : σ ad σ Ukow Examle: Secific Motors Ca we coclude, usig a.05 level of sigificace, that the miles-er-gallo (mg) erformace of M cars is greater tha the miles-ergallo erformace of J cars? Slide 8 4
Hyothesis Tests About µ µ : σ ad σ Ukow Value ad Critical Value Aroaches. Develo the hyotheses. H 0 : µ - µ < 0 H a : µ - µ > 0 where: µ = mea mg for the oulatio of M cars µ = mea mg for the oulatio of J cars Slide 9 Hyothesis Tests About µ µ : σ ad σ Ukow Value ad Critical Value Aroaches. Secify the level of sigificace. α =.05 3. Comute the value of the test statistic. ( x x ) D (9.8 7.3) 0 t = = = 0 = = = s s (.56) (.8) 4 8 4.003 Slide 30 5
Hyothesis Tests About µ µ : σ ad σ Ukow Value Aroach 4. Comute the value. The degrees of freedom for t α are: (.56) (.8) 4 8 df = = 4.07 = 4 (.56) (.8) 4 4 8 8 Because t = 4.003 > t.005 =.797, the value <.005. Slide 3 Hyothesis Tests About µ µ : σ ad σ Ukow Value Aroach 5. Determie whether to reject H 0. Because value < α =.05, we reject H 0. We are at least 95% cofidet that the miles-ergallo (mg) erformace of M cars is greater tha the miles-er-gallo erformace of J cars?. Slide 3 6
Hyothesis Tests About µ µ : σ ad σ Ukow Critical Value Aroach 4. Determie the critical value ad rejectio rule. For α =.05 ad df = 4, t.05 =.7 Reject H 0 if t >.7 5. Determie whether to reject H 0. Because 4.003 >.7, we reject H 0. We are at least 95% cofidet that the miles-ergallo (mg) erformace of M cars is greater tha the miles-er-gallo erformace of J cars?. Slide 33 I-class exercise #9 (.406) (cofidece iterval) #0 (.407) (hyothesis testig) Slide 34 7
Ifereces About the Differece Betwee Two Poulatio Meas: Matched Samles With a matched-samle desig each samled item rovides a air of data values. This desig ofte leads to a smaller samlig error tha the ideedet-samle desig because variatio betwee samled items is elimiated as a source of samlig error. Slide 35 Ifereces About the Differece Betwee Two Poulatio Meas: Matched Samles Examle: Exress Deliveries A Chicago-based firm has documets that must be quickly distributed to district offices throughout the U.S. The firm must decide betwee two delivery services, UPX (Uited Parcel Exress) ad INTEX (Iteratioal Exress), to trasort its documets. Slide 36 8
Ifereces About the Differece Betwee Two Poulatio Meas: Matched Samles Examle: Exress Deliveries I testig the delivery times of the two services, the firm set two reorts to a radom samle of its district offices with oe reort carried by UPX ad the other reort carried by INTEX. Do the data o the ext slide idicate a differece i mea delivery times for the two services? Use a.05 level of sigificace. Slide 37 Ifereces About the Differece Betwee Two Poulatio Meas: Matched Samles District Office Seattle Los Ageles Bosto Clevelad New York Housto Atlata St. Louis Milwaukee Dever Delivery Time (Hours) UPX INTEX Differece 3 30 9 6 5 8 4 0 7 6 5 4 5 5 3 5 5 8 9 7 6 4 3 - - 5 Slide 38 9
Ifereces About the Differece Betwee Two Poulatio Meas: Matched Samles Value ad Critical Value Aroaches. Develo the hyotheses. H 0 : µ d = 0 H a : µ d 0 Let µ d = the mea of the differece values for the two delivery services for the oulatio of district offices Slide 39 Ifereces About the Differece Betwee Two Poulatio Meas: Matched Samles Value ad Critical Value Aroaches. Secify the level of sigificace. α =.05 3. Comute the value of the test statistic. di d= ( 7 6... 5) = = 7. 0 s d = ( d i d ) 76. = = 9. 9 d µ d.7 0 t = = = s.9 0 d.94 Slide 40 0
Ifereces About the Differece Betwee Two Poulatio Meas: Matched Samles Value Aroach 4. Comute the value. For t =.94 ad df = 9, the value is betwee.0 ad.0. (This is a two-tailed test, so we double the uer-tail areas of.0 ad.005.) 5. Determie whether to reject H 0. Because value < α =.05, we reject H 0. We are at least 95% cofidet that there is a differece i mea delivery times for the two services? Slide 4 Ifereces About the Differece Betwee Two Poulatio Meas: Matched Samles Critical Value Aroach 4. Determie the critical value ad rejectio rule. For α =.05 ad df = 9, t.05 =.6. Reject H 0 if t >.6 5. Determie whether to reject H 0. Because t =.94 >.6, we reject H 0. We are at least 95% cofidet that there is a differece i mea delivery times for the two services? Slide 4
I-class exercise # (.44) (hyothesis testig) # (.44) (cofidece iterval) Slide 43 Samlig Distributio of Exected Value E ( ) = Stadard Deviatio (Stadard Error) ( ) ( ) σ = where: = size of samle take from oulatio = size of samle take from oulatio Slide 44
Samlig Distributio of If the samle sizes are large, the samlig distributio of ca be aroximated by a ormal robability distributio. The samle sizes are sufficietly large if all of these coditios are met: > 5 ( - ) > 5 > 5 ( - ) > 5 Slide 45 Samlig Distributio of ( ) ( ) σ = Slide 46 3
Iterval Estimatio of - Iterval Estimate ( ) ( ) ± z α / Slide 47 Iterval Estimatio of - Examle: Market Research Associates Market Research Associates is coductig research to evaluate the effectiveess of a cliet s ew advertisig camaig. Before the ew camaig bega, a telehoe survey of 50 households i the test market area showed 60 households aware of the cliet s roduct. The ew camaig has bee iitiated with TV ad ewsaer advertisemets ruig for three weeks. Slide 48 4
Iterval Estimatio of - Examle: Market Research Associates A survey coducted immediately after the ew camaig showed 0 of 50 households aware of the cliet s roduct. Does the data suort the ositio that the advertisig camaig has rovided a icreased awareess of the cliet s roduct? Slide 49 Poit Estimator of the Differece Betwee Two Poulatio Proortios = roortio of the oulatio of households aware of the roduct after the ew camaig = roortio of the oulatio of households aware of the roduct before the ew camaig = samle roortio of households aware of the roduct after the ew camaig = samle roortio of households aware of the roduct before the ew camaig 0 60 = =.48.40 =.08 50 50 Slide 50 5
Iterval Estimatio of - For α =.05, z.05 =.96:.48(.5).40(.60).48.40 ±.96 50 50.08.96(.050).08.0 Hece, the 95% cofidece iterval for the differece i before ad after awareess of the roduct is -.0 to.8. Slide 5 Hyothesis Tests about - Hyotheses We focus o tests ivolvig o differece betwee the two oulatio roortios (i.e. = ) H 0 : 0 H : < a 0 H - < 0 : 0 H : - a > 0 H 0 : = 0 H : 0 a Left-tailed Right-tailed Two-tailed Slide 5 6
Hyothesis Tests about - Pooled Estimate of Stadard Error of σ = ( ) where: = Slide 53 Hyothesis Tests about - Test Statistic z = ( ) ( ) Slide 54 7
Hyothesis Tests about - Examle: Market Research Associates Ca we coclude, usig a.05 level of sigificace, that the roortio of households aware of the cliet s roduct icreased after the ew advertisig camaig? Slide 55 Hyothesis Tests about - -Value ad Critical Value Aroaches. Develo the hyotheses. H 0 : - < 0 H a : - > 0 = roortio of the oulatio of households aware of the roduct after the ew camaig = roortio of the oulatio of households aware of the roduct before the ew camaig Slide 56 8
Hyothesis Tests about - -Value ad Critical Value Aroaches. Secify the level of sigificace. α =.05 3. Comute the value of the test statistic. 50 (. 48 ) 50 (. 40 ) 80 = = =. 45 50 50 400 s 45 55 =. (. )( ) 50 =. 054 50 (.48.40) 0.08 z = = =.054.054.56 Slide 57 Hyothesis Tests about - Value Aroach 4. Comute the value. For z =.56, the value =.0594 5. Determie whether to reject H 0. Because value > α =.05, we caot reject H 0. We caot coclude that the roortio of households aware of the cliet s roduct icreased after the ew camaig. Slide 58 9
Hyothesis Tests about - Critical Value Aroach 4. Determie the critical value ad rejectio rule. For α =.05, z.05 =.645 Reject H 0 if z >.645 5. Determie whether to reject H 0. Because.56 <.645, we caot reject H 0. We caot coclude that the roortio of households aware of the cliet s roduct icreased after the ew camaig. Slide 59 I-class exercise #30 (.40) (cofidece iterval) #3 (.4) (hyothesis testig) Slide 60 30
Ed of Chater 0 Slide 6 Formulas for two oulatio case Cofidece Iterval Test Statistics Other formula Meas: σ kow Meas: σ ukow σ σ x x ± zα / s s x x ± tα / ( x x ) D z = σ σ 0 ( x x ) D t = s s 0 s s df = s s Meas: matched Same as oe oulatio case Proortio ( ) ( ) ± zα / z = ( ) ( ) = Slide 6 3