9.2. Section. Chapter. Objectives. Estimating the Value of a Parameter Using. The Logic in Constructing. for a Population Mean. Confidence Intervals

Transcription

1 Chapter 9 Etimatig the Value of a Parameter Uig Cofidece Iterval Sectio 9. The Logic i Cotructig Cofidece Iterval for a Populatio Mea 010 Pearo Pretice Hall. All right reerved 010 Pearo Pretice Hall. All right reerved Objective 1. Kow the propertie of Studet t-ditributio. Determie t-value 3. Cotruct ad iterpret a cofidece iterval for a populatio mea Objective 1 I thi ectio, we will try to fid 90%, 95%, ad 99% cofidece iterval for µ give a radom ample. I previou ectio, we ued the tadardized ormal curve (z-core), which work woderfully if the value of σ i kow. Sice we do ot kow the value of µ, we will ot kow the value of σ. So we will ue the bet et thig,. But thi mea we will have to ue aother type of bellhaped curve, the Studet t-ditributio curve. 010 Pearo Pretice Hall. All right reerved Pearo Pretice Hall. All right reerved 9-4 The Studet t- ditributio curve i bell haped like the tadardized ormal curve. It i appropriate to ue the t-ditributio curve if the populatio tadard deviatio i ukow ad the ize of the ample i mall. If i large, the two curve are roughly the ame. The t-ditributio i horter ad more pread out. The total area uder both curve i 1. Parallel Eample: Comparig the Stadard Normal Ditributio to the Studet t-ditributio Uig Simulatio a) Obtai 1,000 imple radom ample of ize =5 from a ormal populatio with =50 ad =10. b) Determie ad for each ample. Note: will chage for each ample of ize 5. It varie where doe ot; it will tay 10. Thu, the t-ditributio i much more comple. c) Compute z t ad for each ample. Thi tadardize the curve; zero will be i the ceter for both curve d) Draw a hitogram for both z ad t. 010 Pearo Pretice Hall. All right reerved Pearo Pretice Hall. All right reerved 9-6 1

2 Hitogram for z (tall ad kiy) Hitogram for t (horter ad fatter) 010 Pearo Pretice Hall. All right reerved Pearo Pretice Hall. All right reerved 9-8 t i called the t-value or critical value for alpha. It i the tadardized value of t o that the area to it right i equal to alpha. Sice varie for the Studet t- ditributio, there i additioal parameter aociated with thi ditributio, degree of freedom. Degree of freedom i equal to 1. Eample: Fidig t-value Thu, if were 0 ad alpha 0.05, the t-value may be deoted t 0.05,19 Calculator t 0.05,19 = ivt(1-.05,19) = 1.79 Note: ivt(.05,19) = Fid the t-value uch that the area uder the t- ditributio to the right of the t-value i 0. ad i 11. That i, fid t 0.0 with 10 degree of freedom. 010 Pearo Pretice Hall. All right reerved Pearo Pretice Hall. All right reerved 9-10 Objective Fidig a cofidece iterval for the populatio mea To get a cofidece iterval, you will eed 1. The poit etimate. The margi of error A poit etimate i the value of a tatitic that etimate the value of a parameter. For eample, the ample mea, the populatio mea., i a poit etimate of Parallel Eample 1: Computig a Poit Etimate Peie mited after 198 are made from 97.5% zic ad.5% copper. The followig data repreet the weight (i gram) of 17 radomly elected peie mited after Treat the data a a imple radom ample. The ditributio of the weight of peie i ormal; thu, the ample ize doe ot have to be at leat 30 for the ample mea to be ormally ditributed. Etimate the populatio mea weight of peie mited after Pearo Pretice Hall. All right reerved Pearo Pretice Hall. All right reerved 9-1

3 The ample mea i Solutio The poit etimate of i.464 gram. I objective, we try to fid the margi of error E o that we are 99% cofidet that the populatio mea i betwee.464 E ad.464+e. Criteria The hape of the ditributio of all poible ample mea will be bell-haped, provided the populatio i ormal or approimately ormal or if the ample ize i large ( 30). (if we fid oe, we foud the other) tadard error i (thi i almot the tadard deviatio of the ample mea) The margi of error i t 010 Pearo Pretice Hall. All right reerved Pearo Pretice Hall. All right reerved 9-14 Cotructig a (1- ) 100% Cofidece Iterval for Suppoe that a imple radom ample of ize i take from a populatio with ukow mea,. A (1- ) 100% cofidece iterval for i give by Lower Boud: t Why i alpha plit i half? Upper Boud: t If alpha i 5%, the we are lookig for a 95% cofidece iterval..5% i i each tail. Area ot i the tail i equal to 95%. Iterpretatio of a Cofidece Iterval A (1- ) 100% cofidece iterval idicate that, if we obtaied may imple radom ample of ize from the populatio whoe mea,, i ukow, the approimately (1- ) 100% of the iterval will cotai. = the area i both tail. /=area i oe tail. 010 Pearo Pretice Hall. All right reerved Pearo Pretice Hall. All right reerved 9-16 Parallel Eample 3: Cotructig a Cofidece Iterval Cotruct a 99% cofidece iterval about the populatio mea weight (i gram) of peie mited after t.91, Lower boud: t = = =.451 Upper boud: t = =.477 We are 99% cofidet that the mea weight of peie mited after 198 i betwee.451 ad.477gram. 010 Pearo Pretice Hall. All right reerved Pearo Pretice Hall. All right reerved

4 Calculator Objective 3 Udertad the Role of the Margi of Error i Cotructig a Cofidece Iterval 010 Pearo Pretice Hall. All right reerved Pearo Pretice Hall. All right reerved 9-0 The margi of error, E, i a (1- ) 100% cofidece iterval i give by E t where i the ample ize. Parallel Eample 5: Role of the Level of Cofidece i the Margi of Error Cotruct a 90% cofidece iterval for the mea weight of peie mited after 198. Commet o the effect that decreaig the level of cofidece ha o the margi of error. Note: We require that the populatio from which the ample wa draw be ormally ditributed or the ample ize be greater tha or equal to Pearo Pretice Hall. All right reerved Pearo Pretice Hall. All right reerved 9- t Lower boud: t = = = Upper boud: t = =.471 We are 90% cofidet that the mea weight of peie mited after 198 i betwee.457 ad.471gram. 010 Pearo Pretice Hall. All right reerved 9-3 Notice that the margi of error decreaed from 0.01 to whe the level of cofidece decreaed from 99% to 90%. The iterval i therefore wider for the higher level of cofidece. Cofidece Level Margi of Error 90% % Pearo Pretice Hall. All right reerved 9-4 4

5 99% Cofidece Iterval Notice that the margi of error decreaed from to whe the ample ize icreaed from 17 to 35. Sample Size Margi of Error Sectio Cofidece Iterval for a Populatio Proportio Pearo Pretice Hall. All right reerved Pearo Pretice Hall. All right reerved Objective 1. Obtai a poit etimate for the populatio proportio. Cotruct ad iterpret a cofidece iterval for the populatio proportio 3. Determie the ample ize eceary for etimatig a populatio proportio withi a pecified margi of error Objective 1 Obtai a poit etimate for the populatio proportio 010 Pearo Pretice Hall. All right reerved Pearo Pretice Hall. All right reerved 9-8 A poit etimate i a ubiaed etimator of the parameter. The poit etimate for the populatio proportio i p ˆ where i the umber of idividual i the ample with the pecified characteritic ad i the ample ize. Parallel Eample 1: Calculatig a Poit Etimate for the Populatio Proportio I July of 008, a Quiipiac Uiverity Poll aked 1783 regitered voter atiowide whether they favored or oppoed the death pealty for pero covicted of murder. 113 were i favor. Obtai a poit etimate for the proportio of regitered voter atiowide who are i favor of the death pealty for pero covicted of murder. 010 Pearo Pretice Hall. All right reerved Pearo Pretice Hall. All right reerved

6 Solutio Obtai a poit etimate for the proportio of regitered voter atiowide who are i favor of the death pealty for pero covicted of murder. Objective Cotruct ad Iterpret a Cofidece Iterval for the Populatio Proportio ˆ p Pearo Pretice Hall. All right reerved Pearo Pretice Hall. All right reerved 9-3 For a imple radom ample of ize, the amplig ditributio of p ˆ i almot ormal provided 0.05N, ˆ p (1 p ˆ ) 10 The mea i The tadard deviatio of the amplig proportio (tadard error) i p ˆ Samplig Ditributio of ˆ p p(1 p) p ˆ p 010 Pearo Pretice Hall. All right reerved 9-33 Cotructig a (1- ) 100% Cofidece Iterval for a Populatio Proportio Suppoe that a imple radom ample of ize i take from a populatio. A (1- ) 100% cofidece iterval for p i give by the followig quatitie Lower boud: p ˆ (1 p ˆ z Upper boud: p ˆ (1 p ˆ z Note: It mut be the cae that ˆ p (1 0.05N to cotruct thi iterval. ˆ p ) ˆ p ) ˆ p ) 10 ad 010 Pearo Pretice Hall. All right reerved 9-34 Parallel Eample : Cotructig a Cofidece Iterval for a Populatio Proportio I July of 008, a Quiipiac Uiverity Poll aked 1783 regitered voter atiowide whether they favored or oppoed the death pealty for pero covicted of murder. 113 were i favor. Obtai a 90% cofidece iterval for the proportio of regitered voter atiowide who are i favor of the death pealty for pero covicted of murder. p ˆ 0.63 ˆ p (1 p ˆ ) 1783(0.63)(1 0.63) ad the ample ize i defiitely le tha 5% of the populatio ize =0.10 o z / =z 0.05 =1.645 Lower boud: Upper boud: Solutio (1 0.63) (1 0.63) Pearo Pretice Hall. All right reerved Pearo Pretice Hall. All right reerved

7 Solutio We are 90% cofidet that the proportio of regitered voter who are i favor of the death pealty for thoe covicted of murder i betwee 0.61ad Pearo Pretice Hall. All right reerved Pearo Pretice Hall. All right reerved 9-38 Objective 3 Determie the Sample Size Neceary for Etimatig a Populatio Proportio withi a Specified Margi of Error Sample ize eeded for a pecified margi of error, E, ad level of cofidece (1- ): ˆ p (1 p ˆ ) z E Problem: The formula ue p ˆ which deped o, the quatity we are tryig to determie! 010 Pearo Pretice Hall. All right reerved Pearo Pretice Hall. All right reerved 9-40 Two poible olutio: 1. Ue a etimate of p baed o a pilot tudy or a earlier tudy.. Let p ˆ =0.5 which give the larget poible value of for a give level of cofidece ad a give margi of error. The ample ize required to obtai a (1- ) 100% cofidece iterval for p with a margi of error E i give by ˆ p (1 p ˆ ) z E (rouded up to the et iteger), where p ˆ i a prior etimate of p. If a prior etimate of p i uavailable, the ample ize required i 0.5 z E 010 Pearo Pretice Hall. All right reerved Pearo Pretice Hall. All right reerved 9-4 7

8 Parallel Eample 4: Determiig Sample Size A ociologit wated to determie the percetage of reidet of America that oly peak Eglih at home. What ize ample hould be obtaied if he wihe her etimate to be withi 3 percetage poit with 90% cofidece aumig he ue the 000 etimate obtaied from the Ceu 000 Supplemetary Survey of 8.4%? E=0.03 z z p ˆ 0.84 Solutio 0.84(1 0.84) We roud thi value up to 437. The ociologit mut urvey 437 radomly elected America reidet. 010 Pearo Pretice Hall. All right reerved Pearo Pretice Hall. All right reerved 9-44 E=0.03 z z pˆ 0.5 Solutio if ot uig etimate (1 0.5) We roud thi value up to 75. The ociologit mut urvey 75 radomly elected America reidet. 010 Pearo Pretice Hall. All right reerved