CH19 Cofidece Itervals for Proportios Cofidece itervals Costruct cofidece itervals for populatio proportios
Motivatio
Motivatio We are iterested i the populatio proportio who support Mr. Obama. This sample has sample size 600 ad sample proportio 47%. What ca we say about the populatio proportio from this sample? Ca we say the populatio proportio is 47%? 51%? 43%? Or betwee 43% ad 51%?
Motivatio Clarificatio: The populatio proportio is a costat. The value of the populatio proportio is fixed. We do ot kow this value. The sample proportio is radom. Differet samples may result differet values of sample proportio. We ca compute sample proportio from a sample.
Review: samplig distributio of sample proportio The sample proportio is a radom variable. The samplig distributio of the sample proportio ^pof samples of size is approximately Normalwith mea p ad stadard deviatio p(1 p) provided that the sampled values are idepedet (e.g. a simple radom sample from a large populatio) ad the sample size is large eough,
Review: samplig distributio of sample proportio mea( = p, ad SD( = p(1 p). p 3SD( p 2SD( pˆ ) p SD( p p SD ( pˆ ) p 2SD( pˆ ) p 3SD( pˆ ) ^ The probability that p beig 2 sd from p is 95%
Cofidece iterval The probability that pis withi 2 stadard deviatio from pis 95%. Same as ^ The probability that p 2SD( pˆ ) pˆ p+ 2SD( pˆ ) is 95%. Same as The probability that p ˆ 2SD( pˆ ) p pˆ + is 95%. 2SD( pˆ )
Cofidece iterval Cautios: The populatio proportio p is ot radom. The iterval ( p ˆ 2SD( pˆ ), pˆ + 2SD( pˆ )) is radom. With 95% chace, this radom iterval will cotai the populatio proportio p.
Cofidece iterval Problem The radom iterval ( p ˆ 2SD( pˆ ), pˆ + 2SD( pˆ )) still depeds o psice p(1 p) SD( =. Remedy: use stadard error to replace stadard deviatio of ^p: pˆ(1 SE( =.
Cetral Limit Theorem for sample proportio The samplig distributio of the sample proportio pof samples of size is approximately Normal with mea p ad stadard deviatio pˆ(1, provided that the sampled values are idepedet (e.g. a simple radom sample from a large populatio) ad the sample size is large eough ^
Cofidece iterval The 95% cofidece iterval for populatio proportio p costructed from samples of size with sample proportio ^pis give by pˆ 2 pˆ(1, pˆ + 2 pˆ(1 For example, a 95% cofidece iterval for the populatio proportio of supportig Mr. Obama (=600, ^p=47%) is (0.4292, 0.5108).
Meaig of cofidece What do we mea by 95%? Or more specifically, what do we mea by sayig we have 95% cofidece that the iterval (0.4292, 0.5108) cotais the true populatio proportio? The populatio proportio is a costat, ot radom. So is the iterval (0.4292, 0.5108). The the probability that the populatio proportio is i (0.4292, 0.5108) is 0 or 1.
Meaig of cofidece The iterval is radom. pˆ 2 pˆ(1, pˆ(1 The actual iterval varies sample by sample. Amog all possible itervals costructed this way, 95% of them will cotai the populatio proportio p. pˆ + 2
Meaig of cofidece Amog all possible itervals costructed this way, 95% of them will cotai the populatio proportio p.
Meaig of cofidece Whe we say we have 95% cofidece that the iterval (0.4292, 0.5108) cotais the true populatio proportio, we mea that we are 95% cofidet the iterval (0.4292, 0.5108) is oe of those itervals which cotais the true populatio proportio.
Margi of error The 95% cofidece iterval for p is pˆ 2 This is same as pˆ(1 pˆ ± 2, pˆ pˆ(1 + 2 pˆ(1 The extet of the iterval o either side of pis called the margi of error (ME). Margi of Error ^
Margi of error Margi of error measures our tolerace for error. It is closely related to cofidece level. For example, based o the CNN poll with =600, ^p=47%, which of the followig statemets are you more cofidet? The populatio proportio pof supportig Mr. Obama is betwee 0 ad 100%; pis betwee 43% ad 51%; pis betwee 45% ad 49%; pis betwee 46.9% ad 47.1%. Margi of error icreases as cofidece level icreases.
Margi of error The margi of error correspodig to 95% cofidece level is 2SE(p). ^ E.g. For the CNN poll, the stadard error is pˆ(1 0.47(1 0.47) SE ( = = = 0.0204, 600 ad the margi of error correspodig to 95% cofidece level is 2(0.0204)=0.0408. What is the margi of error correspodig to 99% cofidece level?
Margi of error ad critical values For cofidece itervals of populatio proportio, the margi of error equals z*se(p), ^ where z* is determied by the cofidece level C. z* is called the critical value correspodig to cofidece level C. Youmayfidz* withti usigbelowcommad: For C=(1-α)%, ivnorm( 1-α/2, 0,1) For example C=95%, α=0.05, 1-α/2=0.975, ivnorm( 0.975, 0,1)=1.96 2 C=90%, α=0.10, 1-α/2=0.95, ivnorm( 0.95, 0,1)=1.645 C=99%, α=0.01, 1-α/2=0.995, ivnorm( 0.995, 0,1)=2.575
Cofidece itervals I geeral the level C cofidece iterval for populatio proportio pis give by pˆ z * pˆ(1, pˆ + z * pˆ(1 where ^pis the sample proportio, is the sample size, ad z* is the critical value correspodig to the cofidece level C.
Cofidece itervals We may also use ˆ± ME p to deote the level C cofidece iterval for populatio proportio p, where ^pis the sample proportio ad ME is the margi of error: ME= z pˆ(1 * pˆ )
Cofidece itervals Practice A CNN poll shows that 47% of 600 surveyed Ohio voters will support Mr. Obama. We are iterested i the populatio proportio pof all Ohio voters who will support Mr. Obama. Costruct a 90% cofidece iterval for p. Costruct a 96% cofidece iterval for p. Costruct a 98% cofidece iterval for p.
Sample size Sample size for desired margi of error: The level C cofidece iterval for a populatio proportio pwill have margi of error approximately equal to a specified value mwhe the sample size is 2 * z = m p * (1 where p* is a guessed value for the sample proportio. The margi of error will be at most m if p* is take to be 0.5. p * )
Practice A isurace compay checks police records o 582 accidets selected at radom ad otes that teeagers were at the wheel i 91 of them. Create a 95% cofidece iterval for the percetage of all auto accidets that ivolve teeage drivers. Explai what 95% cofidece meas. A politicia urgig tighter restrictios o drivers liceses issued to tees says, I oe of every five auto accidets, a teeager is behid the wheel. Does our cofidece iterval support or cotradict this statemet? If this politicia wat to have a 95% cofidece iterval with margi of error 0.10, how large a sample should he survey?
Assumptios ad coditios Idepedece Assumptio: the sampled values must be idepedet of each other Radomizatio Coditio: a radom sample 10% Coditio: the sample size must be o larger tha 10% of the populatio Sample Size Assumptio: the sample size must be large eough Success/Failure Coditio: the sample size has to be big eough so that we expect at least 10 successes ad at least 10 failures.
Suggested exercises from the textbook: Ch19 7, 9, 11, 13, 21, 23, 25, 29, 31, 35, 37, 39, 41