Chapter 7. , and is unknown and n 30 then X ~ t n

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1 Chpter 7 Sectio 7. t-ditributio ( 3) Summry: C.L.T. : If the rdom mple of ize 3 come from ukow popultio with me d S.D. where i kow or ukow, the X ~ N,. Note: The hypothei tetig d cofidece itervl re built uig Z-ditributio. Quetio: Wht i the ditributio of X, if the popultio i ukow d the mple ize i le th 3;( 3)? Awer: Uder thi ce we do't kow the ditributio of the mple me X. Thi ce c be hdled uder No-prmetric ttitic. Quetio: Wht i the ditributio of X, if the popultio i Norml, meig X ~ N, but i Ukow d the mple ize i le th 3; ( 3)? Awer: If the popultio i Norml; X ~ N,, d i ukow d 3 the X ~ t (t-ditributio) with - degree of freedom. Note: The hypothei tetig d cofidece itervl re built uig t- ditributio. The t-ditributio i fmily of curve. The curve look like the tdrd orml ditributio but re ot the me. Ech curve i idetified by the umber of degree of freedom. A the degree of freedom get lrger the t-ditributio i pprochig the tdrd orml curve.

2 Itervl Etimtio of Popultio Me. (Smll Smple Ce, 3) Cofidece Itervl: X t ; Note: Sice i ukow we ued S. Exmple : Aume tht the durtio time of log-ditce telephoe cll i Normlly ditributed. A mple of telephoe cll reulted i mple me of X =.5 miute d mple tdrd devitio of =4. miute. Develop 95% d 98% cofidece itervl etimte of the me durtio time for the popultio of log-ditce telephoe cll. () 95% C.I., - =.95, =.5, =.5, t t =.93 -; 4. X t.5 ; , 4.4. Thu, we re 95% cofidet tht the true popultio me i.58 to 4.4 miute per telephoe cll. 9;.5 (b) 98% C.I., - =.98, =., =., t t -; 9;.= X t.5 ; , 4.83 Thu, we re 98% cofidet tht the true popultio me i.7 to 4.83 miute per telephoe cll.

3 3 ypothei Tetig Geerl form of ypothei Tetig Step Ce. : Ce. : Ce3. : : : : Step X X X t t t Step 3 Re ject if t t ; Re ject if t t ; ject if t t Step 4 Cocluio: We do ot reject d coclude tht... We reject d coclude tht... Re ; Exmple : The height of prticulr plt i Normlly ditributed with me of 8 iche. A ew plt food i ued o mple of plt. Reult of the mple how mple me height of 9.4 iche d mple tdrd devitio of 3 iche. Uig =., i there reo to believe tht the ew plt food icree plt growth?. : 8 v : 8 Upper Til Tet X Tet Sttitic: t t ject if t t ; t t;. Re Cocluio: Sice t.6.363, we reject d coclude tht the me plt height exceed 8 iche whe the ew plt food i ued. Exmple 3: It i etimted tht houewife with hubd d two childre work verge of 55 hour or more per week o houehold relted ctivitie. Show below re the hour worked durig week for mple of eight houewive: 58, 5, 64, 63, 59, 6, 6, 55. Ue =.5 to do the pproprite hypothei tetig if the wive wt to prove to their hubd tht they work more th 55 hour week.

4 4 Begi by computig the followig: X = d S = 4... : 55 v : 55 Upper Til Tet X Tet Sttitic: t t ject if t t ; t7 t7;.5 Re Cocluio: Sice t , we reject d coclude tht the me umber of hour worked per week exceed 55. omework: 7.5, 7. (, d), 7.3, 7.3, 7.4, 7.44 pp Plee Note: ) Uder the t-ditributio do ot compute the p-vlue eve if the problem k for it. If it give to you ue it. ) I thi homework if 3 ue the C.L.T.; i.e. X ~ N,.

5 5 Sectio 7. Comprig Two Popultio Me I thi ectio, we will compre the me of two popultio. For exmple,. Doe Advil fight pi better th Tyleol? b. Doe Loghor Stekhoue hve loger wit th Outbck Stekhoue? c. Are me older th wome whe they grdute from college? d. Are SAT core higher fter tkig preprtio coure th before? e. Are the trtig lrie for computer ciece mjor higher th thoe of mrketig mjor? C.L.T. : If rdom mple of ize d ( 3 d 3) re drw from two popultio with me d d with tdrd devitio of d, repectively, the the mplig ditributio of X X h the followig propertie:. E( X - X ) = -. S.D( X - X )= X = X 3. X - X i pproximtely ormlly ditributed. Tht i, X - X ~ N( -, ) Note:. A C.I. for - h the form: ( X - X ) Z. If d re ukow, we will ue S d S to etimte d, repectively. S S Tht i the C. I. i ( X - X ) Z Itervl Etimtio of : Lrge-Smple Ce. ( 3 d 3 S A % C.I. for i XXZ or X X Z Note: If d re ukow, the we c ue the mple S.D., d. ) S Exmple : The Eductiol Tetig Service coducted tudy to ivetigte poible differece betwee the core of mle d femle o the choltic Aptitude Tet (SAT) (Jourl of Eductiol Meuremet, Sprig 987). A rdom mple of 56 femle d 85 mle provided mple me SAT verbl core of X =547 for the femle d X =55 for the mle. The mple tdrd devitio were =83 for the femle d =78 for the mle. Uig 95% cofidece level, etimte the differece betwee the me SAT verbl core for the two popultio.

6 95% C.I., - =.95, =.5, =.5, Z Z =.96 S S XX Z , 3.63 We c be 95% cofidet tht the me verbl core of femle i bout 3.37 to 3.63 poit higher th the me verbl core for mle. Exmple : I mple of 6 foreig imilr ize cr the verge g milege w X =4 d =9. I mple of 8 dometic imilr ize cr the verge g milege w X =35 d =. Give 9% d 99% C.I. to etimte the differece i g milege betwee foreig d dometic cr. Fid 9% d 99% cofidece itervl We c be 9% cofidet tht the me g milege of the foreig cr i.35 to 7.65 gllo higher th the me g milege of the dometic cr. 9% (4-35) (.645)(.65) 5.65 (.35 to 7.65) We c be 99% cofidet tht the me g milege of the foreig cr i.85 to 9.5 gllo higher th the me g milege of the dometic cr. A hould be expected the 99% C.I. i wider th the 95% C.I.. 99% (4-35) (.575)(.65) (.85 to 9.5).5 6

7 Itervl Etimtio of : Smll-Smple Ce. ( 3 d 3) Sice 3 d 3 the mplig ditributio of X X h to be t- ditributio. We hve two ce to coider Ce ( 3 d 3). Both popultio mut hve Norml ditributio; i.e. X ~ N, d X ~ N,. The tdrd devitio of the two popultio mut be equl; i.e.,, where i the tdrd devitio for both popultio. Agi, d re ukow. X X ~ t The X X. NOTE: The vrice,, will be ukow. We etimte uig wht we cll the pooled vrice, p. p ece, X X p. NOTE: Sice X X ~ t, the C.I. i give by X Xt ; p where p Exmple 3. A urvey of recet grdute with degree i Mrketig d Accoutig reveled the followig trtig lrie i thoud of dollr. Mrketig: 4.6, 7.8, 5.8, 3., 5., 4.7, 6.,.6, 3.8 ( 9, X 4.86 d.6) Accoutig: 6.7, 4.9, 7., 3., 7.5, 7.4, 4.9, 6.3, 8.9, 6.4, 8., 9.7, 5.5, 4.9, 8.5 ( 5, X 6.66 d.79) Build 96% C.I. for the me differece i the trtig lrie of Accoutig d Mrketig mjor? p X Xt ; p ,. We re 96% cofidet tht the Mrketig lrie re $ to $3,39 lower th Accoutig. 7

8 8 Ce ( 3 d 3). Both popultio mut hve Norml ditributio; i.e. X ~ N, d X ~ N,. X X ~ t k where k mi, d X X Cofidece Itervl X X t NOTE: Sice, the C.I. i give by ~ k k; X X t Exmple 4: Coider the followig iformtio bout the verge cr peed i tow for the teeger, mle d femle. Mle :, X 5.48,. Femle : 3, X 4.5, 7.5 Build 95% C.I. for the differece of the two popultio me. k ; ;.5 3 X X t t , 8.9 We re 95% cofidet tht the mle drive i tow to 8.9 mile fter th the femle.

9 ypothei Tetig for the differece Betwee the Me of Two Popultio ) Lrge Smple Ce. ( 3 d 3) Geerl form of ypothei Tetig Step Ce. : Ce. : Ce3. : : : : X X X X X X Step Step 3 Z Re ject if Z Z Z Re ject if Z Z Z Re ject if Z Step 4 Cocluio: We do ot reject d coclude tht... We reject d coclude tht... Z NOTE: The P-Vlue i other wy of drwig cocluio. The P-Vlue i the probbility of obtiig the tet ttitic uder the umptio tht the Null ypothei i true ( i true). If the P Vlue, the we filed to reject. If the P Vlue, the we reject. Tht i, if the probbility of obtiig the tet ttitic i greter th or equl to, X i ot oe of the extreme vlue but oe tht i cloe to. * * Compute the P Vlue : Let Z Tet Sttitic; Tht i Z Z X X * * * Ce: P Vlue P Z Z Ce: P Vlue P Z Z Ce3: P Vlue P Z Z 9 Exmple : A medicl reerch tudy w coducted to determie if there i differece betwee the effectivee of two pi relief medicie ued for hedche. Over 6-moth period, mple of idividul oe of the medicie, where other mple of idividul ued the other medicie. Dt collected durig the tudy howed the time required to receive pi relief.

10 Smple Size =48 =5 Smple Me X =4.8 X =6. Smple S.D. =3.3 =4. Let =me pi relief time for medicie Let =me pi relief time for medicie. : v : Two Til Tet. 3. X X Z Re ject if Z Z Z Cocluio: Sice Z = 3.7 i greter th.96, we reject d coclude tht there i igifict differece betwee the me pi relief time for the two medicie. * P Z Z = P(Z < -3.7) = ()= Note: P-Vlue = P(Z < Z * )= P-Vlue = < =.5. Therefore, we reject d drw the me cocluio i tep 4 bove. Exmple : It h bee uggeted tht college tudet ler more d obti higher grde i mll cle (4 tudet or le) compred to lrge cle (5 tudet or more). To tet thi clim, uiverity iged profeor to tech mll cl d lrge cl of the me coure. At the ed of the coure tudet from the two cle were give the me fil exm. Fil grde differece for the two cle would provide bi for tetig the differece betwee the mll cl d lrge cl itutio. Lettig deote the me exm core for the popultio of tudet tkig mll cl d deote the me exm core for the popultio of tudet tkig lrge cl. Smple Size =35 =7 Smple Me X =74. X =7.7 Smple S.D. =4 =3

11 ) Smll-Smple Ce ( 3 d 3) Geerl form of ypothei Tetig Ce. : Ce. : Ce3. : : : : XX t where p p Re ject if t t ; Re ject if t t ; ject if t t Step Step Step 3. : v : Upper Til Tet. X X Z ject if Z Z Z.5 Re Cocluio: Sice Z =.97 i le th.645, we fil to reject d coclude tht there i o evidece tht tudet i mll cle ler more. * Note: P-Vlue = PZ Z = -P(Z <.97) =-.834=.66 P-Vlue =.66 > =.5. Therefore, we fil to reject d drw the me cocluio i tep 4 bove. Step 4 Cocluio: We do ot reject d coclude tht... We reject d coclude tht... Re ; Exmple 3: Automobile golie milege tet were coducted for imilr ized foreig d dometic utomobile. Tet the hypothei tht the me umber of mile per gllo i the me for foreig d dometic utomobile bed o the followig mple reult. Ue =.5. Aume tht the utomobile golie milege i ormlly ditributed. Smple Size =8 = Smple Me X =36.5 X =3.4 Smple S.D. =.3 =.8

12 Step : v : XX t p where p Step Step 3 Re ject if t t t t. 6 6;.5 ; Step 4 Cocluio: Sice 3.33 >., We reject d coclude tht there i igifict differece betwee the verge golie milege of the foreig d dometic cr. omework: 7.57, 7.69(b, c, d oly), 7.74(c oly), 7.76(b, c oly), 7.78(, b), 7.8, 7.83 pp NOTE: Ue the Z-ditributio whe you c; i.e 3 d 3. Alo: If the exercie k for the p-vlue, do ot compute it uder the t-ditributio. You c compute it if you re uig the Z-ditributio.