Estimating Proportions with Confidence
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- Maurice Cross
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1 Aoucemets: Discussio today is review for midterm, o credit. You may atted more tha oe discussio sectio. Brig sheets of otes ad calculator to midterm. We will provide Scatro form. Homework: (Due Wed Chapter 0: #5,, 4 Chapter 0 Estimatig Proportios with Cofidece Cofidece iterval example from Fri lecture Gallup poll of = 08 adults foud 39% believe i evolutio. So =.39 A 95% cofidece iterval or iterval estimate for the proportio (or percet of all adults who believe i evolutio is.36 to.4 (or 36% to 4%. Cofidece iterval: a iterval of estimates that is likely to capture the populatio value. Goal today: Lear to calculate ad iterpret cofidece itervals for p ad for p p ad lear geeral format. Remember populatio versus sample: Populatio proportio: the fractio of the populatio that has a certai trait/characteristic or the probability of success i a biomial experimet deoted by p. The value of the parameter p is ot kow. Sample proportio: the fractio of the sample that has a certai trait/characteristic deoted by. The statistic is a estimate of p. The Fudametal Rule for Usig Data for Iferece: Available data ca be used to make ifereces about a much larger group if the data ca be cosidered to be represetative with regard to the questio(s of iterest. 3 4 Some Defiitios: Poit estimate: A sigle umber used to estimate a populatio parameter. For our five situatios: poit estimate = sample statistic = sample estimate = for oe proportio = for differece i two proportios Iterval estimate: A iterval of values used to estimate a populatio parameter. Also called a cofidece iterval. For our five situatios, always: Sample estimate ± multiplier stadard error Details for proportios: Sample estimate ± multiplier stadard error Parameter Sample estimate Stadard error p p p p ˆ s. e.( = ( See p. 44 for formula 5 6
2 Multiplier ad Cofidece Level More about the Multiplier The multiplier is determied by the desired cofidece level. The cofidece level is the probability that the procedure used to determie the iterval will provide a iterval that icludes the populatio parameter. Most commo is.95. If we cosider all possible radomly selected samples of same size from a populatio, the cofidece level is the fractio or percet of those samples for which the cofidece iterval icludes the populatio parameter. See picture o board. Ofte express the cofidece level as a percet. Commo levels are 90%, 95%, 98%, ad 99%. Note: Icrease cofidece level => larger multiplier. Multiplier, deoted as z*, is the stadardized score such that the area betwee z* ad z* uder the stadard ormal curve correspods to the desired cofidece level. 7 8 Formula for C.I. for proportio Sample estimate ± multiplier stadard error For oe proportio: A cofidece iterval for a populatio proportio p, based o a sample of size from that populatio, with sample proportio is: ± z * ( Example of differet cofidece levels Poll o belief i evolutio: = 08 Sample proportio =.39 Stadard error = (.39(.39 = = % cofidece iterval.39 ±.65(.053 or.39 ±.05 or.365 to.45 95% cofidece iterval:.39 ± (.053 or.39 ±.03 or.36 to.4 99% cofidece iterval.39 ±.58(.053 or.39 ±.04 or.35 to Iterpretatio of the cofidece iterval ad cofidece level: We are 90% cofidet that the proportio of all adults i the US who believe i evolutio is betwee.365 ad.45. We are 95% cofidet that the proportio of all adults i the US who believe i evolutio is betwee.36 ad.4. We are 99% cofidet that the proportio of all adults i the US who believe i evolutio is betwee.35 ad.43. Iterpretig the cofidece level of 99%: The iterval.35 to.43 may or may ot capture the true proportio of adult Americas who believe i evolutio But, i the log ru this procedure will produce itervals that capture the ukow populatio values about 99% of the time. So, we are 99% cofidet that it worked this time. Notes about iterval width Higher cofidece <=> wider iterval Larger (sample size <=> more arrow iterval, because is i the deomiator of the stadard error. So, if you wat a more arrow iterval you ca either reduce your cofidece, or icrease your sample size.
3 Recocilig with Chapter 3 formula for 95% cofidece iterval Sample estimate ± Margi of error where (coservative margi of error was Now, margi of error is p ˆ( These are the same whe =.5. The ew margi of error is smaller for ay other value of So we say the old versio is coservative. It will give a wider iterval. 3 Comparig three versios (Details o board For the evolutio example, = 08, Coservative margi of error = Approximate margi of error usig z* =.053 = Exact margi of error usig z* = = All very close to.03, ad it really does t make much differece which oe we use! =.39 4 New example: compare methods Marist Poll i Oct 009 asked How ofte do you text while drivig? = 06 Nie percet aswered Ofte or sometimes so ad =.09.09(.9 s. e.( = = Coservative margi of error =.03 Approximate margi of error =.009 =.08. This time, they are quite differet! The coservative oe is too coservative, it s double the approximate oe! 5 Comparig margi of error Coservative margi of error will be okay for sample proportios ear.5. For sample proportios far from.5, closer to 0 or, do t use the coservative margi of error. Resultig iterval is wider tha eeded. Note that usig a multiplier of is called the approximate margi of error; the exact oe uses multiplier of.96. It will rarely matter if we use istead of Factors that Determie Margi of Error. The sample size,. Whe sample size icreases, margi of error decreases.. The sample proportio,. If the proportio is close to either or 0 most idividuals have the same trait or opiio, so there is little atural variability ad the margi of error is smaller tha if the proportio is ear The multiplier or.96. Coected to the 95% aspect of the margi of error. Usually the term margi of error is used oly whe the cofidece level is 95%. Geeral Descriptio of the Approximate 95% CI for a Proportio Approximate 95% CI for the populatio proportio: ± stadard errors ( The stadard error is s. e.( = Iterpretatio: For about 95% of all radomly selected samples from the populatio, the cofidece iterval computed i this maer captures the populatio proportio. Necessary Coditios: ˆ p ad ( are both greater tha 0, ad the sample is radomly selected. 7 8
4 Fidig the formula for a 95% CI for a Proportio use Empirical Rule: For 95% of all samples, is withi st.dev. of p Samplig distributio of tells us for 95% of all samples: stadard deviatios < ˆp p < stadard deviatios Do t kow true stadard deviatio, so use stadard error. For approximately 95% of all samples, stadard errors < ˆp p < stadard errors which implies for approximately 95% of all samples, stadard errors < p < + stadard errors Same holds for ay cofidece level; replace with z* ( ± z where: is the sample proportio z* deotes the multiplier.. ( is the stadard error of. 9 0 Example 0.3 Itelliget Life Elsewhere? Poll: Radom sample of 935 Americas Do you thik there is itelliget life o other plaets? Results: 60% of the sample said yes, =.60.6 ( (.6 s. e. = = % Cofidece Iterval:.60 ±.65(.06, or.60 ± to.66 or 57.4% to 6.6% 98% Cofidece Iterval:.60 ±.33(.06, or.60 ± to.637 or 56.3% to 63.7% Note: etire iterval is above 50% => high cofidece that a majority believe there is itelliget life. Cofidece itervals ad plausible values Remember that a cofidece iterval is a iterval estimate for a populatio parameter. Therefore, ay value that is covered by the cofidece iterval is a plausible value for the parameter. Values ot covered by the iterval are still possible, but ot very likely (depedig o the cofidece level. Example of plausible values 98% Cofidece iterval for proportio who believe itelliget life exists elsewhere is:.563 to.637 or 56.3% to 63.7% Therefore, 56% is a plausible value for the populatio percet, but 50% is ot very likely to be the populatio percet. Etire iterval is above 50% => high cofidece that a majority believe there is itelliget life. New multiplier: let s do a cofidece level of 50% Poll: Radom sample of 935 Americas Do you thik there is itelliget life o other plaets? Results: 60% of the sample said yes, =.60 We wat a 50% cofidece iterval. If the area betwee -z* ad z* is.50, the the area to the left of z* is.75. From Table A. we have z*.67. (See ext page for Table A. 50% Cofidece Iterval:.60 ±.67(.06, or.60 ± to.6 or 58.9% to 6.% Note: Lower cofidece level results i a arrower iterval. 3 4
5 Remember coditios for usig the formula:. Sample is radomly selected from the populatio. Note: Available data ca be used to make ifereces about a much larger group if the data ca be cosidered to be represetative with regard to the questio(s of iterest.. Normal curve approximatio to the distributio of possible sample proportios assumes a large sample size. Both ˆ p ad ( should be at least 0 (although some say these eed oly to be at least 5. I Summary: Cofidece Iterval for a Populatio Proportio p Geeral CI for p: Approximate 95% CI for p: z ± ± Coservative ± 95% CI for p: ( ( 5 6 Sectio 0.4: Comparig two populatio proportios Idepedet samples of size ad Use the two sample proportios as data. Could compute separate cofidece itervals for the two populatio proportios ad see if they overlap. Better to fid a cofidece iterval for the differece i the two populatio proportios, 7 Case Study 0.3 Comparig proportios Would you date someoe with a great persoality eve though you did ot fid them attractive? Wome:.6 (6.% of 3 aswered yes. 95% cofidece iterval is.57 to.694. Me:.46 (4.6% of 6 aswered yes. 95% cofidece iterval is.30 to.55. Coclusios: Higher proportio of wome would say yes. CIs slightly overlap Wome CI arrower tha me CI due to larger sample size 8 Compare the two proportios by fidig a CI for the differece C.I. for the differece i two populatio proportios: Sample estimate ± multiplier stadard error ( ± z * ( ˆ ˆ p( p + Case Study 0.3 Comparig proportios Would you date someoe with a great persoality eve though you did ot fid them attractive? Wome:.6 of 3 aswered yes. 95% cofidece iterval is.57 to.694. Me:.46 of 6 aswered yes. 95% cofidece iterval is.30 to.55. Cofidece iterval for the differece i populatio proportios of wome ad me who would say yes. (.6.46 ± z *.6(.6.46(
6 95% cofidece iterval A 95% cofidece iterval for the differece is.035 to.334 or 3.5% to 33.4%. We are 95% cofidet that the populatio proportios of me ad wome who would date someoe they did t fid attractive differ by betwee.035 ad.334, with a lower proportio for me tha for wome. We ca coclude that the two populatio proportios differ because 0 is ot i the iterval. Sectio 0.5: Usig cofidece itervals to guide decisios A value ot i a cofidece iterval ca be rejected as a likely value for the populatio parameter. Whe a cofidece iterval for p p does ot cover 0 it is reasoable to coclude that the two populatio values differ. Whe cofidece itervals for p ad p do ot overlap it is reasoable to coclude they differ, but if they do overlap, o coclusio ca be made. I that case, fid a cofidece iterval for the differece. 3 3 From the Midterm review sheet for Chapter 0 - you should kow these ow. Uderstad how to iterpret the cofidece level. Uderstad how to iterpret a cofidece iterval 3. Uderstad how the samplig distributio for leads to the cofidece iterval formula (pg Kow how to compute a cofidece iterval for oe proportio, icludig coditios eeded. 5. Kow how to compute a cofidece iterval for the differece i two proportios, icludig coditios eeded. 6. Uderstad how to fid the multiplier for desired cofidece level. 7. Uderstad how margi of error from Chapter 3 relates to the 95% cofidece iterval formula i Chapter 0 8. Kow the geeral format for a cofidece iterval for the 5 situatios defied i Chapter 9 (see summary o page
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