Confidence Intervals (2) QMET103

Transcription

1 Cofidece Iterval () QMET03 Library, Teachig ad Learig

2 Geeral Remember: three value are ued to cotruct all cofidece iterval: Samle tatitic Z or t Stadard error of amle tatitic Deciio ad Parameter to idetify: Whether differece are betwee mea or betwee roortio Whether amle are ideedet or ot For each ideedet amle, the mea ad tadard deviatio, or the roortio Whether the give tadard deviatio (or variace) are from oulatio or amle The level of cofidece required For t core, the ize of each amle ad hece degree of freedom:

3 IDENTIFY the amle tatitic For two amle cofidece iterval, thi i either mea differece, x : calculate the differece d betwee the air of data, ad roce thee differece a oe amle OR differece betwee mea, x x : calculate the mea of the two amle ad calculate the differece betwee thee two mea OR differece betwee roortio, calculate the differece betwee the two roortio SELECT the relevat Z or t core The value of t deed o the level of cofidece ad the amle ize. : It i ued whe the tadard deviatio i from a amle ie i ukow The value of Z deed o the level of cofidece oly. Ue it o whe the tadard deviatio i from a oulatio ie i kow, or o for roortio 3

4 CALCULATE the tadard error You will fid thee formulae o your formula heet. For differece betwee MEANS d e(mea differece), e x : d From the roceed differece, ue the.d of the differece divided by the quare root of the amle ize e(differece betwee mea), oulatio d kow: e x x e(differece betwee mea), oulatio d NOT kow (amle d kow): e x x where ooled var iace. Thi mut be calculated firt, the ued to calculate the tadard error. For differece betwee PROPORTIONS e(differece betwee roortio), for INDEPENDENT amle: e or x x e where e(differece betwee roortio), for NON INDEPENDENT amle: e 4

5 Examle Two deedet amle, reult aired A Iurace Comay obtaied etimate of the cot of car reair at a certai garage. The iurace comay radomly elected five car eedig reair ad obtaied the actual cot of the fiihed reair. The data are below: Car Etimate ($) Actual ($) From thee aired reult, the differece for each air i obtaied ad roceed: Differece Thi i ow treated a a igle amle, with x % Cofidece iterval ow = = [$-59.8, -$.98] 5 (No cocluio, ice iterval cotai zero.) Two ideedet amle, with oulatio read give or kow That i, amle mea ad oulatio tadard deviatio or variace are kow. Weekly exediture of tudet of two Uiveritie are tudied. Variace of the two oulatio are 90. ad 97.7, reectively. From oulatio, a amle of 5 tudet i elected with amle mea beig $04.0. From oulatio, a amle of 0 tudet i elected ad the mea i $ What i the 95% Cofidece Iterval for the differece betwee the two oulatio mea? Idetify: x 04. 0, x , 90., 97. 7, 5, 0 C.I.(differece betwee mea)= x x Z $.8, $ 7.39,ice i kow. That i, we ca be 95% cofidet that the true mea differece i exediture betwee the two uiveritie i betwee $.8 ad $7.39. Note that i thi examle, the variace were give, ot the tadard deviatio. There i o eed therefore to quare 90. ad 97.7 i the.e. formula. Two ideedet amle, with amle read give or kow That i, amle mea ad amle tadard deviatio or variace are kow. 0 male tudet were radomly amled to determie the mea umber of hour er week male ed watchig televiio. The mea for thi amle wa 5.3 hour ad the tadard deviatio 4. hour. A imilar urvey of 5 female roduced a mea of. hour ad a tadard deviatio of 3.0 hour. Calculate the 90% cofidece iterval for the differece i hour et watchig televiio betwee the male ad female grou. Iterret thi cofidece iterval. Idetify: x 5. 3, x., 4., 3. 0, 0, 5 5

6 C.I.(diff betwee mea)= x x t, ice i ukow Firt calculate.8 43 Now comlete the C.I , That i, we ca be 95% cofidet that the true mea differece i hour et watchig televiio betwee male ad female i betwee.3 ad 5.9 hour. Two ideedet amle, roortio give. ie, two earate amle. The eior accout maager of a chai of ho i cocered about the differece betwee ho i their cotrol of a tock hrikage (lo due to theft ad oor ivetory cotrol). She thik that the differece i due to the tye of hoig cetre the ho are located i: either i mall or with treet frotage. She collect the followig data. Tye of locality Mall Street frotage Number of ho 8 0 Stock hrikage % % Calculate the 95% cofidece iterval for the differece i hrikage betwee the ho tye Idetify: Hece mall treet. C. I ad 0. 0, the robability of ucce from each amle 8, 0. the ize of each amle Z - core C. I ,0. Becaue there i a zero i the 95% C.I., (the iterval goe form oitive to egative), o cocluio ca be draw. Thi imlie that the differece i tock hrikage could be zero. Two No- ideedet amle, roortio give; ometime referred to a a ame amle urvey; ice the roortio come from a igle amle. Surveyor aked a radom amle of 0 wome, what factor wa the mot imortat i decidig where to ho. The reult are ummaried i the followig table: Factor Percetage (%) Price ad value 40 Quality ad electio of merchadie 30 Service 5 Shoig eviromet 5

7 Calculate a 95% cofidece iterval for the differece betwee the oulatio roortio of wome who idetified rice ad value a mot imortat ad the roortio of wome who idetified quality ad electio of merchadie a mot imortat ( rice ad value - quality ad electio ) I thi ituatio, the roortio i each category i iter deedet. A chage i oe value will affect oe or more of the other. Idetify: 0. 4 ad 0. 3, the robability of ucce from each amle Note, there i oly oe amle ize. the ize of the amle, 0 C.I. Z ( ) , 0. 5 Agai, there i a zero i the 95% C.I., o o cocluio ca be draw. Thi imlie that the differece i referece could be zero. Practice. A real-etate comay araied the market value of 7 home i Lyttelto ad foud that the amle mea ad tadard deviatio were $50,000 ad $7,000 reectively. The real-etate comay alo araied the market value of 45 home i Araui ad foud that the amle mea ad tadard deviatio were $00,000 ad $,000 reectively. Calculate the 90% cofidece iterval etimate for the oulatio differece i market value betwee the Lyttelto ad Araui area ( Lyttelto - Araui ). [5 Mark]. The Coutry Tate bread makig comay wat to etimate the actual weight of their 700 gm bread. It i kow that the govermet ecificatio for the tadard deviatio of weight of thi bread i 5 gm. A radom amle of 50 bread i elected, ad the amle mea weight i 696 gm. Aother comay, South Tate, alo coducted a imilar tudy with a radom amle of 60 bread ad foud that the amle mea i 70 gm. Calculate the 90% cofidece iterval etimate for the differece betwee the two oulatio mea weight coutry outh. 3. A amle of 50 obervatio well are radomly elected from Caterbury regio ad it i foud that itrate level i 3 well exceed the allowable limit. Of amle of 60 well i Waikato area 8% of the well idicate that their itrate level i over the accetable limit. Cotruct a 90% cofidece iterval for the differece betwee the two oulatio roortio that exceed the allowable itrate cocetratio. 7

8 4. I a Rugby World Cu, a radom amle of uorter wa aked, Which coutry do you thik will wi the 003 Rugby World Cu? The reult are ummaried: Coutry Number of uorter who thik their coutry will wi Autralia 6 Eglad 3 Frace 5 New Zealad 40 Weter Samoa 50 South Africa 47 Wale 4 Udecided 65 Total 480 Calculate the 95% cofidece iterval for the differece betwee the roortio who thik Autralia will wi ad the roortio who thik that New Zealad will wi. 5. A farmer wat to examie the effect a ew drech i havig o the weight of hi hee. To do thi, he weigh 0 hee rior to drechig ad the reweigh the ame hee week after drechig. The meauremet obtaied are give i the table below. Shee o. Weight before (kg) Weight after (kg) Calculate a 95% cofidece iterval for the differece i hee weight after drechig. 6. I a tudy, a grou of 4 edetary me were laced o a diet. After 6 moth thee me had lot a average of 7. kg of bodyweight with a tadard deviatio of 3.7 kg. a) Calculate the 95% cofidece iterval etimate of the oulatio mea weight lo. b) I a earate tudy, 47 reviouly edetary me were ut o a exercie routie. After 6 moth, thee me had lot a average of 4.0kg with a tadard deviatio of 3.9kg. Calculate the 95% cofidece iterval etimate for the oulatio differece i mea weight lo ( diet - exercie ) c) Iterret the cofidece iterval calculated i (b). I a tudy ivetigatig the bet treatmet to hel eole to to mokig, 44 moker were give Zyba (atidereat). After 6 moth 85 of the 44 Zyba uer had quit mokig. d) Calculate the 95% cofidece iterval etimate of the oulatio roortio who had quit mokig. 8

9 I additio to the Zyba treatmet, a further 44 moker were give icotie atche. After 6 moth, 5 of thee atch uer had give u mokig. e) Calculate the 95% cofidece iterval etimate for the differece i the oulatio roortio of eole givig u mokig uig the differet treatmet ( Zyba - Patche ). Awer C.I. x x t , , $ , $ We ca be 90% cofidet the true mea differece i market value betwee the Lyttelto ad Araui area i betwee $44,000 ad $56,000 5 x 696, x 70, 50, 60 C. I. x x Z (0.75, 9. 7) We ca be 90% cofidet the true differece betwee the two oulatio mea weight coutry outh i betwee 0.75 ad 9.7gm , Thee are ideedet amle with 50, C.I

10 No cocluio ca be made about the oulatio differece i roortio exceedig itrate level for Waikato ad Caterbury. 4. Thee roortio are o-ideedet, with P = 0.47; P = 0.97 C. I , 0.5 Z That i, we ca be 95% cofidet that the oulatio differece i roortio of thoe who thik Autralia will wi the ext world cu ad thoe who thik NZ will wi the ext world cu. 5. Differece i hee after drechig are: Mea differece ad.d. are. ad.83 reectively. Hece, 6. 4, x 7.kg, 3. 7 a) For 95% C.I., t =.005 b). 83 C.I , C.I.( mea ) For a 95% C.I. df = 87, ad t = , kg 4, x 7. kg, 3. 7kg diet diet diet 47, x 4. 0kg, 3. 9kg exercie exercie exercie Pooled variace eeded = C.I.( differece betwee mea ) , kg c) We ca be 95% cofidet that the true oulatio mea differece i weight lo i betwee 0.78 ad 5.6kg 0

11 d) For 95% C.I., Z=.96 C.I.( rortio ) , e) 0.348, 0. 3 Zyba atch For 95% C.I., Z=.96 C. I.( diff. roortio ) ,

12 CONFIDENCE INTERVAL FOR MEAN Oe Oe amle or two? Two No Do you kow the oulatio read? No Matched Pair? Ye Fid differece betwee air C.I. x t Ye Pool variace No Do you kow the oulatio read? C. I. Ye x x Z No d C.I. xd t Do you kow the oulatio read? Ye C.I. x Z ad ue: C.I. x x t C.I. x d d Z CONFIDENCE INTERVAL FOR PROPORTION Oe Oe amle or two? Two Ideedet or Same Samle Survey? C.I. Z Ideedet Same amle urvey C.I. Z C.I. Z