Chapter, Part A Aalyss of Varace ad Epermetal Desg Itroducto to Aalyss of Varace Aalyss of Varace: Testg for the Equalty of Populato Meas Multple Comparso Procedures Itroducto to Aalyss of Varace Aalyss of Varace (ANOVA ca be used to test for the equalty of three or more populato meas. Data obtaed from observatoal or epermetal studes ca be used for the aalyss. We wat to use the sample results to test the followg hypotheses: H 0 :... H a : Not all populato meas are equal Slde Slde Itroducto to Aalyss of Varace H 0 :... H a : Not all populato meas are equal If H 0 s reected, we caot coclude that all populato meas are dfferet. Reectg H 0 meas that at least two populato meas have dfferet values. Itroducto to Aalyss of Varace Samplg Dstrbuto of Gve H 0 s True Sample meas are close together because there s oly oe samplg dstrbuto whe H 0 s true. σ σ Slde Slde 4 Itroducto to Aalyss of Varace Samplg Dstrbuto of Gve H 0 s False Sample meas come from dfferet samplg dstrbutos ad are ot as close together whe H 0 s false. Assumptos for Aalyss of Varace For each populato, the respose varable s ormally dstrbuted. The varace of the respose varable, deoted σ, s the same for all of the populatos. The observatos must be depedet. Slde 5 Slde 6
Aalyss of Varace: Testg for the Equalty of Populato Meas Betwee-Treatmets Estmate of Populato Varace Wth-Treatmets Estmate of Populato Varace Comparg the Varace Estmates: The F Test ANOVA Table Betwee-Treatmets Estmate of Populato Varace A betwee-treatmet estmate of σ s called the mea square treatmet ad s deoted MSTR. MSTR ( Deomator represets the degrees of freedom assocated wth SSTR Numerator s the sum of squares due to treatmets ad s deoted SSTR Slde 7 Slde 8 Wth-Samples Estmate of Populato Varace The estmate of σ based o the varato of the sample observatos wth each sample s called the mea square error ad s deoted by MSE. MSE Deomator represets the degrees of freedom assocated wth SSE ( s T Numerator s the sum of squares due to error ad s deoted SSE Comparg the Varace Estmates: The F Test If the ull hypothess s true ad the ANOVA assumptos are vald, the samplg dstrbuto of MSTR/MSE s a F dstrbuto wth MSTR d.f. equal to - ad MSE d.f. equal to T -. If the meas of the populatos are ot equal, the value of MSTR/MSE wll be flated because MSTR overestmates σ. Hece, we wll reect H 0 f the resultg value of MSTR/MSE appears to be too large to have bee selected at radom from the approprate F dstrbuto. Slde 9 Slde 0 Hypotheses Test Statstc H 0 :... H a : Not all populato meas are equal F MSTR/MSE Reecto Rule p-value Approach: Crtcal Value Approach: Reect H 0 f p-value < α Reect H 0 f F > F α where the value of F α s based o a F dstrbuto wth - umerator d.f. ad T - deomator d.f. Slde Slde
Samplg Dstrbuto of MSTR/MSE ANOVA Table Reecto Rego Samplg Dstrbuto of MSTR/MSE Reect H 0 Do Not Reect H 0 α F α Crtcal Value MSTR/MSE Source of Varato Treatmet Error Total Sum of SSTR SSE SST SST s parttoed to SSTR ad SSE. Degrees of Freedom T T - Mea MSTR MSE MSTR/MSE SST s degrees of freedom (d.f. are parttoed to SSTR s d.f. ad SSE s d.f. F Slde Slde 4 ANOVA Table ANOVA Table SST dvded by ts degrees of freedom T s the overall sample varace that would be obtaed f we treated the etre set of observatos as oe data set. Wth the etre data set as oe sample, the formula for computg the total sum of squares, SST, s: SST ( SSTR + SSE ANOVA ca be vewed as the process of parttog the total sum of squares ad the degrees of freedom to ther correspodg sources: treatmets ad error. Dvdg the sum of squares by the approprate degrees of freedom provdes the varace estmates ad the F value used to test the hypothess of equal populato meas. Slde 5 Slde 6 Eample: Reed Maufacturg Jaet Reed would le to ow f there s ay sgfcat dfferece the mea umber of hours wored per wee for the departmet maagers at her three maufacturg plats ( Buffalo, Pttsburgh, ad Detrot. Eample: Reed Maufacturg A smple radom sample of fve maagers from each of the three plats was tae ad the umber of hours wored by each maager for the prevous wee s show o the et slde. Coduct a F test usg α.05. Slde 7 Slde 8
Observato 4 5 Sample Mea Sample Varace Plat Buffalo 48 54 57 54 6 Plat Pttsburgh 7 6 66 64 74 Plat Detrot 5 6 6 54 56 55 68 57 6.0 6.5 4.5 p -Value ad Crtcal Value Approaches. Develop the hypotheses. H 0 : H a : Not all the meas are equal where: mea umber of hours wored per wee by the maagers at Plat mea umber of hours wored per wee by the maagers at Plat mea umber of hours wored per wee by the maagers at Plat Slde 9 Slde 0 p -Value ad Crtcal Value Approaches p -Value ad Crtcal Value Approaches. Specfy the level of sgfcace. α.05. Compute the value of the test statstc. Mea Square Due to Treatmets (Sample szes are all equal. (55 + 68 + 57/ 60 SSTR 5(55-60 + 5(68-60 + 5(57-60 490 MSTR 490/( - 45. Compute the value of the test statstc. Mea Square Due to Error SSE 4(6.0 + 4(6.5 + 4(4.5 08 MSE 08/(5-5.667 F MSTR/MSE 45/5.667 9.55 (cotued Slde Slde ANOVA Table p Value Approach Source of Varato Treatmet Error Total Sum of 490 08 798 Degrees of Freedom 4 Mea 45 5.667 F 9.55 4. Compute the p value. Wth umerator d.f. ad deomator d.f., the p-value s.0 for F 6.9. Therefore, the p-value s less tha.0 for F 9.55. 5. Determe whether to reect H 0. The p-value <.05, so we reect H 0. We have suffcet evdece to coclude that the mea umber of hours wored per wee by departmet maagers s ot the same at all plat. Slde Slde 4 4
Crtcal Value Approach 4. Determe the crtcal value ad reecto rule. Based o a F dstrbuto wth umerator d.f. ad deomator d.f., F.05.89. Reect H 0 f F >.89 5. Determe whether to reect H 0. Ecel Worsheet (showg data A B C D E Observato Buffalo Pttsburgh Detrot 48 7 5 54 6 6 4 57 66 6 5 4 54 64 54 6 5 6 74 56 Because F 9.55 >.89, we reect H 0. We have suffcet evdece to coclude that the mea umber of hours wored per wee by departmet maagers s ot the same at all plat. Slde 5 Slde 6 Ecel s Aova: Sgle Factor Tool Step Select the Tools meu Step Choose the Data Aalyss opto Step Choose Aova: Sgle Factor from the lst of Aalyss Tools Ecel s Dalog Bo cotued Slde 7 Slde 8 Ecel Value Worsheet A B C D E F G 8 Aova: Sgle Factor 9 0 SUMMARY Groups Cout Sum Average Varace Buffalo 5 75 55 6 Pttsburgh 5 40 68 6.5 4 Detrot 5 85 57 4.5 5 6 7 ANOVA 8 Source of Varato SS df MS F P-value F crt 9 Betwee Groups 490 45 9.54545 0.00.8859 0 Wth Groups 08 5.6667 Total 798 4 Slde 9 Multple Comparso Procedures Suppose that aalyss of varace has provded statstcal evdece to reect the ull hypothess of equal populato meas. Fsher s least sgfcat dfferece (LSD procedure ca be used to determe where the dffereces occur. Slde 0 5
Hypotheses H 0 : H a Reecto Rule p-value Approach: Reect H 0 f p-value < α Test Statstc t MSE( + Crtcal Value Approach: Reect H 0 f t < -t a/ or t > t a/ where the value of t a/ s based o a t dstrbuto wth T - degrees of freedom. Slde Slde Hypotheses Test Statstc Reecto Rule where Based o the Test Statstc - 0 H a Reect H 0 f > LSD LSD t / MSE( α + Based o the Test Statstc - Eample: Reed Maufacturg Recall that Jaet Reed wats to ow f there s ay sgfcat dfferece the mea umber of hours wored per wee for the departmet maagers at her three maufacturg plats. Aalyss of varace has provded statstcal evdece to reect the ull hypothess of equal populato meas. Fsher s least sgfcat dfferece (LSD procedure ca be used to determe where the dffereces occur. Slde Slde 4 Based o the Test Statstc - For α.05 ad T - 5 degrees of freedom, t.05.79 LSD t / MSE( α + LSD. 79 5. 667 ( 5 + 5 698. MSE value was computed earler Based o the Test Statstc - LSD for Plats ad Hypotheses (A Reecto Rule Reect H 0 f > 6.98 Test Statstc 55 68 H 0 : H a Cocluso The mea umber of hours wored at Plat s ot equal to the mea umber wored at Plat. Slde 5 Slde 6 6
Based o the Test Statstc - LSD for Plats ad Hypotheses (B Reecto Rule Test Statstc 55 57 Cocluso Reect H 0 f > 6.98 H 0 : H a There s o sgfcat dfferece betwee the mea umber of hours wored at Plat ad the mea umber of hours wored at Plat. Slde 7 Based o the Test Statstc - LSD for Plats ad Hypotheses (C Reecto Rule Test Statstc 68 57 Cocluso Reect H 0 f > 6.98 H 0 : H a The mea umber of hours wored at Plat s ot equal to the mea umber wored at Plat. Slde 8 Type I Error Rates (P.56 Ed of Chapter, Part A The comparsowse Type I error rate α dcates the level of sgfcace assocated wth a sgle parwse comparso. The epermetwse Type I error rate α EW s the probablty of mag a Type I error o at least oe of the ( / parwse comparsos. α EW ( α (-/ The epermetwse Type I error rate gets larger for problems wth more populatos (larger. Boferro adustmet Use a comparsowse error rate equal α EW /C for test C parwse comparso. (C ( / Slde 9 Slde 40 7