Definition. Definition. 72 Estimating a Population Proportion. Definition. Definition


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1 7 stimatig a Populatio Proportio I this sectio we preset methods for usig a sample proportio to estimate the value of a populatio proportio. The sample proportio is the best poit estimate of the populatio proportio. We ca use a sample proportio to costruct a cofidece iterval to estimate the true value of a populatio proportio, ad we should kow how to iterpret such cofidece itervals. We should kow how to fid the sample size ecessary to estimate a populatio proportio. A poit estimate is a sigle value (or poit) used to approximate a populatio parameter. The sample proportio ˆp is the best poit estimate of the populatio proportio p. 11 xample The Pew Research Ceter coducted a survey of 1007 adults ad foud that 85% of them kow what Twitter is. The best poit estimate of p, the populatio proportio, is the sample proportio: p ˆ 0.85 A cofidece iterval (or iterval estimate) is a rage (or a iterval) of values used to estimate the true value of a populatio parameter. A cofidece iterval is sometimes abbreviated as CI. A cofidece level is the probability 1 α (ofte expressed as the equivalet percetage value) that the cofidece iterval actually does cotai the populatio parameter, assumig that the estimatio process is repeated a large umber of times. (The cofidece level is also called degree of cofidece, or the cofidece coefficiet.) Most commo choices are 90%, 95%, or 99%. (α 0.10), (α 0.05), (α 0.01)
2 Iterpretig a Cofidece Iterval Cautio We must be careful to iterpret cofidece itervals correctly. There is a correct iterpretatio ad may differet ad creative icorrect iterpretatios of the cofidece iterval 0.88 < p < Kow the correct iterpretatio of a cofidece iterval. We are 95% cofidet that the iterval from 0.88 to 0.87 actually does cotai the true value of the populatio proportio p. This meas that if we were to select may differet samples of size 1007 ad costruct the correspodig cofidece itervals, 95% of them would actually cotai the value of the populatio proportio p. (Note that i this correct iterpretatio, the level of 95% refers to the success rate of the process beig used to estimate the proportio.) Cofidece itervals ca be used iformally to compare differet data sets, but the overlappig of cofidece itervals should ot be used for makig formal ad fial coclusios about equality of proportios. Usig Cofidece Itervals for Hypothesis Tests A cofidece iterval ca be used to test some claim made about a populatio proportio p. For ow, we do ot yet use a formal method of hypothesis testig, so we simply geerate a cofidece iterval ad make a iformal judgmet based o the result. Critical Values A stadard z score ca be used to distiguish betwee sample statistics that are likely to occur ad those that are ulikely to occur. Such a z score is called a critical value. Critical values are based o the followig observatios: 1. Uder certai coditios, the samplig distributio of sample proportios ca be approximated by a ormal distributio.. A z score associated with a sample proportio has a probability of α/ of fallig i the right tail. Critical Values 3. The z score separatig the righttail regio is commoly deoted by z α/ ad is referred to as a critical value because it is o the borderlie separatig z scores from sample proportios that are likely to occur from those that are ulikely to occur. A critical value is the umber o the borderlie separatig sample statistics that are likely to occur from those that are ulikely to occur. The umber z α/ is a critical value that is a z score with the property that it separates a area of α/ i the right tail of the stadard ormal distributio.
3 Fidig z α/ for a 95% Cofidece Level Commo Critical Values Cofidece Level α Critical Value, z α/ 90% % % z a / z a / Critical Values Whe data from a simple radom sample are used to estimate a populatio proportio p, the margi of error, deoted by, is the maximum likely differece (with probability 1 α, such as 0.95) betwee the observed proportio ˆp ad the true value of the populatio proportio p. 33 Margi of rror for Proportios The margi of error is also called the maximum error of the estimate ad ca be foud by multiplyig the critical value ad the stadard deviatio of the sample proportios: z α Cofidece Iterval for stimatig a Populatio Proportio p Cofidece Iterval for stimatig a Populatio Proportio p p ˆp z α/ populatio proportio sample proportio umber of sample values margi of error z score separatig a area of α/ i the right tail of the stadard ormal distributio. 1. The sample is a simple radom sample.. The coditios for the biomial distributio are satisfied: there is a fixed umber of trials, the trials are idepedet, there are two categories of outcomes, ad the probabilities remai costat for each trial. 3. There are at least 5 successes ad 5 failures.
4 Cofidece Iterval for stimatig a Populatio Proportio p Cofidece Iterval for stimatig a Populatio Proportio p pˆ < p< pˆ + ˆp ± where z α ( pˆ, pˆ + ) RoudOff Rule for Cofidece Iterval stimates of p Roud the cofidece iterval limits for p to three sigificat digits. 44 Procedure for Costructig a Cofidece Iterval for p 1. Verify that the required assumptios are satisfied. (The sample is a simple radom sample, the coditios for the biomial distributio are satisfied, ad the ormal distributio ca be used to approximate the distributio of sample proportios because p 5, ad q 5 are both satisfied.). Refer to Table A ad fid the critical value z α/ that correspods to the desired cofidece level. 3. valuate the margi of error z ˆˆ α pq Procedure for Costructig a Cofidece Iterval for p  cot 4. Usig the value of the calculated margi of error ad the value of the sample proportio, ˆp, fid the values of ˆp ad ˆp+. Substitute those values i the geeral format for the cofidece iterval: pˆ < p< pˆ + 5. Roud the resultig cofidece iterval limits to three sigificat digits. xample I the Chapter Problem we oted that a Pew Research Ceter poll of 1007 radomly selected adults showed that 85% of respodets kow what Twitter is. The sample results are 1007 ad pˆ a. Fid the margi of error that correspods to a 95% cofidece level. b. Fid the 95% cofidece iterval estimate of the populatio proportio p. c. Based o the results, ca we safely coclude that more tha 75% of adults kow what Twitter is? d. Assumig that you are a ewspaper reporter, write a brief statemet that accurately describes the results ad icludes all of the relevat iformatio.
5 xample  Cotiued Requiremet check: simple radom sample; fixed umber of trials, 1007; trials are idepedet; two outcomes per trial; probability remais costat. Note: umber of successes ad failures are both at least 5. a) Use the formula to fid the margi of error. ( 085)( 015).. zα xample  Cotiued b) The 95% cofidece iterval: pˆ < p < pˆ < p < < p < xample  Cotiued c) Based o the cofidece iterval obtaied i part (b), it does appear that more tha 75% of adults kow what Twitter is. Because the limits of 0.88 ad 0.87 are likely to cotai the true populatio proportio, it appears that the populatio proportio is a value greater tha xample  Cotiued d) Here is oe statemet that summarizes the results: 85% of U.S. adults kow what Twitter is. That percetage is based o a Pew Research Ceter poll of 1007 radomly selected adults. I theory, i 95% of such polls, the percetage should differ by o more tha. percetage poits i either directio from the percetage that would be foud by iterviewig all adults i the Uited States. Aalyzig Polls Whe aalyzig polls cosider: 1. The sample should be a simple radom sample, ot a iappropriate sample (such as a volutary respose sample).. The cofidece level should be provided. (It is ofte 95%, but media reports ofte eglect to idetify it.) 3. The sample size should be provided. (It is usually provided by the media, but ot always.) 4. xcept for relatively rare cases, the quality of the poll results depeds o the samplig method ad the size of the sample, but the size of the populatio is usually ot a factor. Cautio Never follow the commo miscoceptio that poll results are ureliable if the sample size is a small percetage of the populatio size. The populatio size is usually ot a factor i determiig the reliability of a poll.
6 Sample Size Suppose we wat to collect sample data i order to estimate some populatio proportio. The questio is how may sample items must be obtaied? Determiig Sample Size z a (solve for by algebra) ( z ) a Sample Size for stimatig Proportio p Whe a estimate of ( z ) Whe o estimate of a ( z a ) 0.5 ˆp is kow: ˆp is kow: 66 RoudOff Rule for Determiig Sample Size If the computed sample size is ot a whole umber, roud the value of up to the ext larger whole umber. xample May compaies are iterested i kowig the percetage of adults who buy clothig olie. How may adults must be surveyed i order to be 95% cofidet that the sample percetage is i error by o more tha three percetage poits? a. Use a recet result from the Cesus Bureau: 66% of adults buy clothig olie. b. Assume that we have o prior iformatio suggestig a possible value of the proportio. a) Use xample  Cotiued pˆ 0.66 ad qˆ 1 pˆ 0.34 α 0.05 so zα ( zα ) ( 1.96) ( 0.66)( 0.34) ( 0.03) To be 95% cofidet that our sample percetage is withi three percetage poits of the true percetage for all adults, we should obtai a simple radom sample of 958 adults.
7 b) Use xample  Cotiued α 0.05 so z α ( zα ) 0.5 ( ) ( 0.03) To be 95% cofidet that our sample percetage is withi three percetage poits of the true percetage for all adults, we should obtai a simple radom sample of 1068 adults. Fidig the Poit stimate ad from a Cofidece Iterval Poit estimate of : (upper cofidece limit) + (lower cofidece limit) ˆp Margi of rror: (upper cofidece limit) (lower cofidece limit) 77
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