11/19/ Chapter 10 Overview. Chapter 10: Two-Sample Inference. + The Big Picture : Inference for Mean Difference Dependent Samples

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1 /9/0 + + Chapter 0 Overview Dicoverig Statitic Eitio Daiel T. Laroe Chapter 0: Two-Sample Iferece 0. Iferece for Mea Differece Depeet Sample 0. Iferece for Two Iepeet Mea 0.3 Iferece for Two Iepeet Proportio 0.4 Iferece for Two Iepeet Staar Deviatio Lecture PowerPoit Slie + The Big Picture : Iferece for Mea Differece Depeet Sample 4 Where we are comig from a where we are heae Thu far, our iferece ha bee limite to oe populatio a oe ample. Here i Chapter, Two-Sample Iferece, we perform iferece o the ifferece i the parameter of two populatio. I Chapter, we will tur to iferece metho for categorical ata, uch a cotigecy table. Ditiguih betwee iepeet ample a epeet ample. Perform hypothei tet for the populatio mea ifferece for epeet ample. Cotruct a iterpret cofiece iterval for the populatio mea ifferece for epeet ample. Ue a t iterval for µ to perform t tet about µ. Iepeet a Depeet Sample Chapter 0 i about two-ample iferece. The type of iferece we apply epe o whether the ata come from iepeet ample or epeet ample. Iepeet Sample a Depeet Sample Two ample are iepeet whe the ubject electe for the firt ample o ot etermie the ubject i the eco ample. Two ample are epeet whe the ubject i the firt ample etermie the ubject i the eco ample. The ata from epeet ample are calle matche-pair or paire ample. 5 Depeet Sample t Tet for the Populatio Mea of the Differece Paire Sample t Tet: Critical Value Metho For matche pair ata take from epeet ample of two populatio, fi the ifferece to prouce a raom ample of the ifferece betwee the populatio. You ca ue the t tet wheever either of the followig coitio i met: The populatio of ifferece i ormal, or The ample ize lower of ifferece bou = i large x Z( / 30). upper bou = x Z / Step : Fi t crit a tate the rejectio rule. Step 3: Calculate t ata : x tata Step 4: State the cocluio a the iterpretatio. 6

2 /9/0 Depeet Sample t Tet for the Populatio Mea of the Differece 7 Depeet Sample t Tet for the Populatio Mea of the Differece 8 Paire Sample t Tet: p-value Metho For matche pair ata take from epeet ample of two populatio, fi the ifferece to prouce a raom ample of the ifferece betwee the populatio. You ca ue the t tet wheever either of the followig coitio i met: The populatio of ifferece i ormal, or The ample ize lower of ifferece bou = i x large Z( / (30). ) Step : State the hypothee upper bou a the = rejectio x Z / rule. Step : Calculate t ata : x tata Step 3: Fi the p-value. Step 4: State the cocluio a the iterpretatio. Depeet Sample t Tet for the Populatio Mea of the Differece 9 t Iterval for the Populatio Mea Differece for Depeet Sample 0 Cofiece Iterval for Populatio Mea Differece µ For matche-pair ata take from epeet ample of two populatio, fi the ifferece to prouce a raom ample of the ifferece betwee the populatio. A 00( )% cofiece iterval for µ, the populatio mea of the ifferece, i: lower bou = x t upper bou = x t / / ( ( ) ) Thi t iterval applie wheever either of the followig i met: The populatio of ifferece i ormal, or The ample ize of ifferece i large ( 30). The t iterval ca alo be writte a: x t / ( ) Give a t cofiece iterval for µ, we may perform two-taile tet by etermiig whether or ot a pecific value of µ fall withi the iterval. + 0.: Iferece for Two Iepeet Mea Perform a iterpret t tet about µ µ uig Welch metho. Compute a iterpret t iterval for µ µ uig Welch metho. Ue cofiece iterval for µ µ to perform two-taile t tet about µ µ. Perform a iterpret t tet a t iterval µ µ uig the poole variace metho. Apply Z tet a Z iterval for µ µ whe σ a σ are kow. Samplig Ditributio of x x Whe raom ample are raw iepeetly from two populatio with populatio mea µ a µ, a either (a) the populatio are ormally itribute or (b) the ample ize are large ( 30), the the quatity t x x approximately follow a t itributio with f equal to the maller of a. Thi t tatitic i calle Welch approximate t.

3 /9/0 3 Welch Hypothei Tet for the Differece i Two Populatio Mea 4 Welch Hypothei Tet for the Differece i Two Populatio Mea Welch Hypothei Tet: Critical Value Metho The hypothei tet applie wheever either: Both populatio are ormally itribute, or Both ample ize are large ( 30). Step : Fi t crit a tate the rejectio rule. lower bou = x Z / Step 3: Calculate t ata : x x upper tatabou = x Z / Welch Hypothei Tet: p-value Metho The hypothei tet applie wheever either: Both populatio are ormally itribute, or Both ample ize are large ( 30). Step : State the hypothee a the rejectio rule. Step : Calculate t ata : lower bou x x= t x Z / ata upper bou = x Z / Step 3: Fi the p-value. t Cofiece Iterval for µ µ Welch Cofiece Iterval for µ µ For two iepeet raom ample take from two populatio with mea µ a µ, a 00( )% cofiece iterval for µ µ i: (x x ) t / Thi t iterval applie wheever either of the followig i met: Both populatio are ormally itribute, or Both ample ize are large ( 30). The margi of error for thi iterval i give by: E t / Give a t cofiece iterval for µ µ, we may perform two-taile tet by etermiig whether or ot a pecific value of µ µ fall withi the iterval. 5 Example a. Cotruct a 95% cofiece iterval for the ifferece i ru core per game i the America League a Natioal League, uig the ata from Example 0.7. μ a μ repreet AL a NL populatio mea ru per game, repectively: 95% cofiece iterval (-0.7, ) b. Determie with 95% cofiece whether the populatio mea umber of ru core per game i the America League iffer igificatly from the populatio mea umber of ru core i the Natioal League. The cofiece iterval i (a) oe cotai 0. That i, 0 lie betwee -0.7 a Therefore, with 95% cofiece, we o ot reject the hypothei that there i o ifferece betwee populatio mea ru core per game i the America a Natioal league. 6 t Iferece Uig Poole Variace : Iferece for Two Iepeet Proportio 8 A alterative metho for t iferece may be applie whe there i reao to believe that the variace of the two populatio are equal. A poole etimate of the commo variace i ue. Poole Etimate of the Commo Variace σ ( ) ( ) poole Perform a iterpret Z tet for p p. Compute a iterpret Z iterval for p p. The coitio for performig t iferece uig poole variace are the ame a for Welch metho, with the aitioal coitio that the populatio variace are equal. Ue Z iterval for p p to perform two-taile Z tet. 3

4 /9/0 Samplig Ditributio of ^ p ^ p 9 Hypothei Tet for the Differece i Two Populatio Proportio 0 Whe two raom ample are raw iepeetly from two populatio, the the quatity Z p ˆ p ˆ p p p q p q ha a approximately taar ormal itributio whe the followig coitio are atifie: the umber of uccee i each ample are greater tha 5 a the umber of failure i each ample are greater tha 5. Hypothei Tet for the Differece i Two Populatio Proportio: Critical Value Metho Suppoe we have two iepeet raom ample take from two populatio with proportio p a p a the umber of uccee a failure i each ample are all greater tha 5. Step : Fi Z crit a lower tate the bou rejectio = x rule. Z / Step 3: Calculate Z upper ata : bou = x Z / (p ˆ ) Z ata poole( poole) where x x. poole Hypothei Tet for the Differece i Two Populatio Proportio Hypothei Tet for the Differece i Two Populatio Proportio: p-value Metho Suppoe we have two iepeet raom ample take from two populatio with proportio p a p a the umber of uccee a failure i each ample are all greater tha 5. Step : State the hypothee a the rejectio rule. Step : Calculate Zlower ata : bou = x Z / Zata upper bou = x Z / poole( poole) where x x. poole Step 3: Fi the p-value. Example Ue the ata from Table 0.6 a the TI-83/84 to tet whether the populatio proportio of teeage boy who pot their lat ame i their olie profile i greater tha the populatio proportio of teeage girl who o o. Ue the critical value metho a level of igificace α = 0.0. There are 00 boy who repoe ye a 300 who repoe o. Niety-ix girl repoe ye a 404 repoe o. All are greater tha 5. H 0 : p p veru H a : p > p Where p a p repreet, repectively, the populatio proportio of boy a girl who pot their lat ame i their olie profile. Sice the tet i right-taile a = 0.0, Table 0.5 give u Z crit =.33. We will reject H 0 if Z ata >.33. Z ata = 7.6. We have eviece to ugget the populatio proportio of boy who ue their lat ame i greater tha the populatio proportio of girl who o o. Z Cofiece Iterval for p p Cofiece Iterval for p p For two iepeet raom ample take from two populatio with proportio p a p, a 00( )% cofiece iterval for p p, i ( p ˆ p ˆ ) Z / p ˆ ˆ q p ˆ ˆ Thi Z iterval applie whe the umber of uccee a failure i each of the ample are at leat 5. The margi of error for thi iterval i give by p ˆ E Z ˆ q / p ˆ ˆ q Give a Z cofiece iterval for p p, we may perform two-taile tet by etermiig whether or ot a pecific value of p p fall withi the iterval. q : Iferece for Two Iepeet Staar Deviatio Decribe the characteritic of the F itributio a the F tet for two populatio taar eviatio. Perform hypothei tet for two populatio taar eviatio uig the critical-value metho. Perform hypothei tet for two populatio taar eviatio uig the p-value metho. 4 4

5 /9/0 The F Ditributio a the F Tet 5 Propertie of the F Curve 6 I Sectio , we were itrouce to iferece metho for comparig two populatio mea or two populatio proportio. Here we lear how to perform hypothei tet regarig two populatio taar eviatio. Thi require a ew tet, calle the F tet. Tet Statitic for the F Tet Suppoe that the two populatio variace are equal, a that we have iepeet raom ample from two ormally itribute populatio. lower bou = x Z / The the tet tatitic for the F tet upper bou = x Z / Fata follow a F itributio with egree of freeom i the umerator a egree of freeom i the eomiator.. The total area uer the F curve equal.. The value of the F raom variable i ever egative; F curve tart at 0, exte iefiitely to the right, a approache but ever meet the horizotal axi. 3. F curve i right-kewe. 4. Differet F curve for each ifferet pair f a f. Fiig F Critical Value Proceure for Fiig F Critical Value for a Area to the Right Suppoe we have a F itributio with f a f egree of freeom. To fi the critical value F crit that ha area to the right of it: Step : Look acro the top of the F table util you fi your f. The go ow that colum util you ee your f o the left. Step : For each f o the left, you will ee a rage of a value from 0.00 to Chooe the row ext to f that ha your value of. The F-value i that row a colum i your value of F crit. Proceure for Fiig F Critical Value for a Area to the Left Step : Switch the value of f a f. Step : Fi F,, uig the F table. Step 3: Fcrit F,, F,, 7 F Tet for Comparig Two Populatio Staar Deviatio F Tet for Comparig Two Populatio Staar Deviatio: Critical Value Metho Suppoe we have two iepeet raom ample take from two ormally itribute populatio. Step : Fi the critical value() a tate the rejectio rule. Step 3: Calculate Flower ata : bou = x Z / Fata upper bou = x Z / 8 F Tet for Comparig Two Populatio Staar Deviatio 9 + Chapter 0 Overview 30 F Tet for Comparig Two Populatio Staar Deviatio: p-value Metho Suppoe we have two iepeet raom ample take from two ormally itribute populatio. Step : State the hypothee a the rejectio rule. Step : Calculate F lower ata : bou F = x Z / ata upper bou = x Z / Step 3: Fi the p-value. 0. Iferece for Mea Differece Depeet Sample 0. Iferece for Two Iepeet Mea 0.3 Iferece for Two Iepeet Proportio 0.4 Iferece for Two Iepeet Staar Deviatio 5