Distribution of sample means. Estimation

Transcription

1 y y y Chater 9 - Iterval Etimatio Sectio 9.4: Iterval Etimatio: Cofidece Iterval for the Poulatio Mea Ditributio of amle mea. Chater 9 - Iterval Etimatio Sectio 9.4: Iterval Etimatio: Cofidece Iterval for the Poulatio Mea Etimatio Stadard error of the mea 68% Poit Etimate: A igle umber which ue amle iformatio to etimate the value of a oulatio arameter. Iterval Etimate: A etimate of the rage of value withi which the oulatio arameter i likely to fall. 68% of the area i withi (±1 ) of the area i withi ( ± 2 ) 99.7% of the area i withi ( ± 3 ) MAT 102 STATISTICS Dr J. Lubowky Page 1 MAT 102 STATISTICS Dr J. Lubowky Page 2 Chater 9 - Iterval Etimatio Coider a Poulatio Etimatio Overview Decribe the Poulatio uig ad or Chater 9 - Iterval Etimatio Cofidece Iterval for the Poulatio Mea Coider a oulatio of -year old me ad wome. Blood Preure: 1 Take reeated amle of ize 100 Ditributio of amle mea i ormal SE take amle 1 ecall: For a ormal ditributio, of the data fall betwee z 1.64 ad z amle mea ( 1 ) MAT 102 STATISTICS Dr J. Lubowky Page 3 MAT 102 STATISTICS Dr J. Lubowky Page 4

2 Chater 9 - Iterval Etimatio Cofidece Iterval for the Poulatio Mea Coider a oulatio of -year old me ad wome. Blood Preure: 1 Take reeated amle of ize 100 Ditributio of amle mea i ormal SE Take reeated amle of 100 from thi oulatio ad lot the amle mea. z 1.64 z 1.64 IvNormal( 0.95 ) 1 The eted a rage of 1.64 above ad below each amle mea. Oo! Chater 9 - Iterval Etimatio Cofidece Iterval for the Poulatio Mea Coider a oulatio of -year old me ad wome. Blood Preure: 1 Take reeated amle of ize 100 Ditributio of amle mea i ormal ( 1. 64)( 2) + ( 1. 64)( 2) z 1.64 z SE Take reeated amle of 100 from thi oulatio ad lot the amle mea. The iterval etimate for ay articular amle i ± ( 1. 64)( SE) For of the amle mea, the rage eteded about the amle mea will iclude the true mea of the oulatio,. Thu i the CONFIDENCE LEVEL. Thi rage of value i called the CONFIDENCE INTEVAL MAT 102 STATISTICS Dr J. Lubowky Page 5 MAT 102 STATISTICS Dr J. Lubowky Page 6 Chater 9 - Iterval Etimatio Cofidece Iterval for the Poulatio Mea Coider a oulatio of -year old me ad wome. Blood Preure: 1 Take reeated amle of ize 100 Ditributio of amle mea i ormal SE Chater 9 - Iterval Etimatio Sectio 9.4: Iterval Etimatio: Cofidece Iterval for the Poulatio Mea Other Cofidece Level SE z 1.64 z (1.64)(2) (1.64)(2) Take reeated amle of 100 from thi oulatio ad lot the amle mea. To determie the oulatio mea,, at a Cofidece Level, take a amle ad determie it mea,. There i a robability that will be i the rage - (1.64)(SE) < < + (1.64)(SE) Cofidece Level z-core % 2.58 z Cofidece Iterval ± 1.64 ± 1.96 ± 2.58 Margi of Error MAT 102 STATISTICS Dr J. Lubowky Page 7 MAT 102 STATISTICS Dr J. Lubowky Page 8

3 Chater 9 - Iterval Etimatio Sectio 9.4: Iterval Etimatio: Cofidece Iterval for the Poulatio Mea To etimate the mea amout overdue i all it deliquet accout, a bak radomly amle 49 accout ad fid the amle mea,, to be $237. Baed o at hitory, the bak ue a tadard deviatio,, of $ Determie the Cofidece Iterval for the mea amout overdue. 2. Determie the Margi of Error For a Cofidece Level, z IvNormal(.975) 1.96 z 1.96 Sice $53 ad 49, 53 the Stadard Error of the Mea Sice the Samle Mea wa $237, the Cofidece Iterval i - z < < + z $237 - ( 1.96)( 7.57 ) < < $237 + ( 1.96)( 7.57 ) $ < < $ $14.84 Margi of Error Chater 9 - Iterval Etimatio Sectio 9.4: Iterval Etimatio: Cofidece Iterval for the Poulatio Mea To reeat the roblem uig the calulator: $237 $53 49 STATS TESTS ZIterval Margi of Error MAT 102 STATISTICS Dr J. Lubowky Page 9 MAT 102 STATISTICS Dr J. Lubowky Page 10 Chater 9 - Iterval Etimatio Sectio 9.4: Iterval Etimatio: Cofidece Iterval for the Poulatio Mea Ditributio of amle mea. i ukow t Ditributio 5% 5% i the amle tadard deviatio (determied from the amle) t Chater 9 - Iterval Etimatio Sectio 9.4: Iterval Etimatio: Cofidece Iterval for the Poulatio Mea Ditributio of amle mea. i ukow To fid the t-core whe you kow the area uder the t-ditributio, you mut ue a t-table. For eamle, to kow the t-core that ecloe of the area i the ceter of the t-ditributio uig a amle ize of 10, To fid, ay the Cofidece Iterval, determie the t-core which ecloe of the amle mea. Oce the t-core ( t )i foud, t Ditributio CONFIDENCE INTEVAL t < < + t with robability 5% t 5% 1.83 MAT 102 STATISTICS Dr J. Lubowky Page 11 MAT 102 STATISTICS Dr J. Lubowky Page 12

4 Chater 9 - Iterval Etimatio Sectio 9.4: Iterval Etimatio: Cofidece Iterval for the Poulatio Mea To fid the t-core whe you kow the area uder the t-ditributio, you mut ue a t-table. To fid t Degree of Freedom ( df ) i imly ( -1) for 10, df 9 t 1.83 Chater 9 - Iterval Etimatio Sectio 9.4: Iterval Etimatio: Cofidece Iterval for the Poulatio Mea The t-ditributio 5% 5% Aedi D Table III Page 734 Cofidece Level 99% Cofidece Iterval t < < + t t < < + t t < < + t 97.5% 97.5% 99.5% 99.5% MAT 102 STATISTICS Dr J. Lubowky Page 13 MAT 102 STATISTICS Dr J. Lubowky Page 14 Chater 9 - Iterval Etimatio Sectio 9.4: Iterval Etimatio: Cofidece Iterval for the Poulatio Mea Page 547 Eamle 8.3 Eglih rofeor etimate the umber of tyig error er age i term aer by collectig a amle of 36 aer. From thi amle, he fid a mea of 4.6 error er age with 1.4. What i the cofidece iterval for all term aer? Samle Std Dev 1.4 Mea Error/Page 4.6 Samle Size 36 df For Cofidece Iterval, - ( t ) to + ( t ) to to Chater 9 - Iterval Etimatio Sectio 9.6: Determiig Samle Size ad Margi of Error For Cofidece Iterval, - t to + t ( ) ( ) to to Samle mea Cofidece Iterval Margi of Error E.394 MAT 102 STATISTICS Dr J. Lubowky Page 15 MAT 102 STATISTICS Dr J. Lubowky Page 16

5 Chater 9 - Iterval Etimatio Iterval Etimatio Uig the Calculator To etimate the Summer traffic acro the Throg Neck Bridge, the DOT amled the traffic o 10 radomly elected Summer day. I thouad of car, the reult were Chater 9 - Iterval Etimatio Sectio 9.5: Iterval Etimatio: Poulatio Proortio I the oulatio there are millio of eublica, Democrat ad Other thouad car er day. Ue the calculator to etimate the mea daily traffic ad the Margi of Error for the daily Summer traffic at a, ad 99% Cofidece Level. STATS TESTS TIterval O D D D O D D D D D O D D D O The roortio of eublica i the oulatio i Number of eublica i the oulatio N Total oulatio N At a Cofidece Level Et Mea Traffic Margi of Error 7.72 If we take a amle of eole from the oulatio, the roortio of eublica i the amle will be Number of eublica i the amle ˆ Samle ize MAT 102 STATISTICS Dr J. Lubowky Page 17 MAT 102 STATISTICS Dr J. Lubowky Page 18 Chater 9 - Iterval Etimatio Sectio 9.5: Iterval Etimatio: Poulatio Proortio (amle roortio) would NOT equal (oulatio roortio). I geeral If we took reeated amle from the oulatio ad determied the roortio from each amle, they would form a ormal ditributio. Std Error of Pro ˆ (1-) ( 1 ) The mea of the ditributio of the roortio i ad the tadard error of the roortio i Chater 9 - Iterval Etimatio Sectio 9.5: Iterval Etimatio: Poulatio Proortio Whe we are etimatig the oulatio roortio,, from the amle roortio,, ˆ we ue to rereet the tadard error of the roortio. ˆ ( 1 - ˆ ) Therefore, the cofidece iterval of a roortio i : ˆ - ( 1.65 ) < < ˆ + ( 1.65 ) : ˆ - ( 1.96 ) < < ˆ + ( 1.96 ) 99%: ˆ - ( 2.58 ) < < ˆ + ( 2.58 ) MAT 102 STATISTICS Dr J. Lubowky Page 19 MAT 102 STATISTICS Dr J. Lubowky Page

6 Chater 9 - Iterval Etimatio Sectio 9.5: Iterval Etimatio: Poulatio Proortio From a emeter cla urvey of 35 tudet, 30 tudet idicated that they believed i God. From thi amle what i the cofidece iterval for the roortio of all tudet at Naau who believe i God. From the amle Samle roortio ˆ believer 30 amle ize Stadard error of the roortio: ˆ ˆ ˆ ( 1 - ) Ue Normal Ditributio for Cofidece Iterval : : 99%: ( ) ˆ - ( 1.65 ) < < ˆ + ( 1.65 ) ˆ - ( 1.96 ) < < ˆ + ( 1.96 ) ˆ - ( 2.58 ) < < ˆ + ( 2.58 ) Chater 9 - Iterval Etimatio Sectio 9.5: Iterval Etimatio: Poulatio Proortio So ˆ believer ad.059 amle ize the for a cofidece level of : : ˆ - ( 1.65 ) to ˆ + ( 1.65 ) (1.65)(.059) to (1.65)(.059) to : 99%: ˆ - ( 1.96 ) to ˆ + ( 1.96 ) ˆ ˆ (1.96)(.059) to (1.96)(.059) to ˆ - ( 2.56 ) to ˆ + ( 2.56 ) ˆ ˆ (2.56)(.059) to (2.56)(.059) to MAT 102 STATISTICS Dr J. Lubowky Page 21 MAT 102 STATISTICS Dr J. Lubowky Page 22 Chater 9 - Iterval Etimatio Sectio 9.6: Determiig Samle Size ad Margi of Error To etimate the mea flight time betwee two citie, a amle of 64 flight durig the year wa take. The amle had a mea of 2 hour ad a amle tadard deviatio,, of miute. At a cofidece level of, what i the cofidece iterval for all flight time betwee thoe citie? What i the margi of error? Sice i ukow, we ue TIterval with 1 miute, miute, 64 ad C-Level.95 The TIterval i ( 115, 125) miute. So the Margi of Error E ( ) / 2 5 miute Chater 9 - Iterval Etimatio Etimate Po Mea Overview e.g. age, grade, weight, icome, day of getatio, etc. Etimate Po Pro e.g. ro of eublica, ro of 60 year old, ro of female, etc. Take amle of ize ad decide o cofidece level e.g.,, or 99% Determie ad. Determie ˆ /. t-ditributio with df -1 / ˆ 99.5% ( ) ± ( t) where: t t for t t97.5% for t t for 99% ormal ditributio if >5 ad (1-)>5 ± ( z) (1 ) ( ˆ ) where: z 1.65 for z 1.96 for z 2.58 for 99% MAT 102 STATISTICS Dr J. Lubowky Page 23 MAT 102 STATISTICS Dr J. Lubowky Page 24

7 Chater 9 - Iterval Etimatio Chater 9 - Iterval Etimatio Proortio Etimatio Uig the Calculator To etimate a oulatio roortio,, from a amle, ˆ, ue STAT TESTS 1-ProZIt For eamle i a amle of 100 voter, voter aid they would vote the Ideedet cadidate. At a Cofidece level, what roortio of the oulatio of voter are eected to vote Ideedet? Ideedet voter i Samle Samle ize Cofidece Level Etimatio Uig the Calculator Summary Etimate of the Poulatio Mea () kow ZIterval (tat) : o td dev : amle ize : amle mea C-level: Cofidece level Etimate of the Poulatio Proortio () 1-ProZIt : o. of cae i amle : amle ize C-level: Cofidece level The reult, Thu the cofidece iterval for the roortio of voter who we eect will vote Ideedet i betwee.1216 ad <.2 <.2784 Pro Iterval Etimate of the Poulatio Mea () ukow TIterval (tat) : amle td dev : amle ize : amle mea C-level: Cofidece level Margi of Error MAT 102 STATISTICS Dr J. Lubowky Page 25 MAT 102 STATISTICS Dr J. Lubowky Page 26