# Definition. Definition. 7-2 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)

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.

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 Roud-Off 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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