Sectio 9.2 Cofidece Itervals for Proportios We will lear to use a sample to say somethig about the world at large. This process (statistical iferece) is based o our uderstadig of samplig models, ad will be our focus for the rest of the course. I this sectio we leared how to costruct a cofidece iterval for a populatio proportio. Slide 1 Stadard Deviatio Stadard Error Stadard Deviatios for Normal Samplig Distributios are: For proportios For meas _ pq SD pˆ SD( x) Whe we do t kow p or σ, we re stuck, right? Nope. We will use sample statistics to estimate these populatio parameters. Wheever we estimate the stadard deviatio of a samplig distributio, we call it a Stadard Error: Stadard Error for proportios Stadard Error for meas pq ˆˆ _ s SE pˆ SE( x) Slide 2 1
Oe-Proportio z-iterval Whe the coditios are met, we are ready to fid the cofidece iterval for the populatio proportio, p. The cofidece iterval is where ˆp z SE ˆp SE( ˆp) ˆp ˆq The critical value, z*, depeds o the particular cofidece level, C, that you specify. Slide 3 Cofidece iterval for p The Gallup Youth Survey asked a radom sample of 439 U.S. tees aged 13 to 17 whether they thought youg people should wait to have sex util marriage. Of the sample, 246 said Yes. Costruct ad iterpret a 95% cofidece iterval for the proportio of all tees who would say Yes if asked this questio. Check coditios: EXAMPLE: Tees Say Sex Ca Wait Fid the z value: Fid ˆp : Calculate the cofidece iterval: Coclude (i cotext): Slide 4 2
Choosig Your Sample Size The questio of how large a sample to take is a importat step i plaig ay study. Choose a Margi or Error (ME) ad a Cofidece Iterval Level. The formula requires ˆp which we do t have yet because we have ot take the sample. Whe possible do a pilot study. A good estimate for ˆp, which will yield the largest value for (ad ˆp ˆq therefore for ) is 0.50. Solve the formula for. ME z * ˆp ˆq Slide 5 EXAMPLE: Choosig Your Sample Size A compay has received complaits about its customer service. The maagers ited to hire a cosultat to carry out a survey of customers. Before cotactig the cosultat, the compay presidet wats some idea of the sample size that they will be required to pay for. Oe critical questio is the degree of satisfactio with the compay s customer service, measured o a 5-poit scale. The presidet wats to estimate the proportio p of customers who are satisfied (that is, who choose either satisfied or very satisfied, the 2 highest levels o the 5-poit scale). The presidet wats the estimate to be withi 3% (.03) at a 95% cofidece level. How large a sample is eeded? Slide 6 3
Margi of Error: Certaity vs. Precisio Slide 7 Margi of Error: Certaity vs. Precisio (cot.) To be more cofidet, we wid up beig less precise. We eed more values i our cofidece iterval to be more certai. Because of this, every cofidece iterval is a balace betwee certaity ad precisio. The tesio betwee certaity ad precisio is always there. Fortuately, i most cases we ca be both sufficietly certai ad sufficietly precise to make useful statemets. The choice of cofidece level is somewhat arbitrary, but keep i mid this tesio betwee certaity ad precisio whe selectig your cofidece level. The most commoly chose cofidece levels are 90%, 95%, ad 99% (but ay percetage ca be used). Slide 8 4
Key Poit We ve leared to iterpret a cofidece iterval by Tellig what we believe is true i the etire populatio from which we took our radom sample. Of course, we ca t be certai but we ca be cofidet. Review the followig Key Poits for HW See me with questios Slide 9 Slide 10 5
Review Assumptios ad Coditios All statistical models make upo assumptios. Differet models make differet assumptios. If those assumptios are ot true, the model might be iappropriate ad our coclusios based o it may be wrog. You ca ever be sure that a assumptio is true, but you ca ofte decide whether a assumptio is plausible by checkig a related coditio. Here are the assumptios ad the correspodig coditios you must check before creatig a cofidece iterval for a proportio: Idepedece Assumptio: We first eed to Thik about whether the Idepedece Assumptio is plausible. It s ot oe you ca check by lookig at the data. Istead, we check two coditios to decide whether idepedece is reasoable. Radomizatio Coditio: Were the data sampled at radom or geerated from a properly radomized experimet? Proper radomizatio ca help esure idepedece. 10% Coditio: Is the sample size o more tha 10% of the populatio? Normal Coditio for Proportios - Sample Size Assumptio: The sample eeds to be large eough for us to be able to use the CLT ad use a Normal model. Success/Failure Coditio: We must expect at least 10 successes ad at least 10 failures. Slide 11 Review Fidig Critical Values The 2 i pˆ 2 SE( pˆ) (our 95% cofidece iterval) came from the 68-95-99.7% Rule. Usig Table A or techology, we fid that a more exact value for our 95% cofidece iterval is 1.96 istead of 2. We call 1.96 the critical value ad deote it z*. For ay cofidece level, we ca fid the correspodig critical value (the umber of SEs that correspods to our cofidece iterval level). Example: For a 90% cofidece iterval, the critical value is 1.645. How do you fid this umber? Table A or ivnorm(.05,0,1) Slide 12 6
Key Poits: What Ca Go Wrog? Do t Misstate What the Iterval Meas: Do t suggest that the parameter varies. Do t claim that other samples will agree with yours. Do t be certai about the parameter. Do t forget: It s about the parameter (ot the statistic). Do t claim to kow too much. Do take resposibility (for the ucertaity). Do treat the whole iterval equally. Margi of Error Too Large to Be Useful: We ca t be exact, but how precise do we eed to be? Oe way to make the margi of error smaller is to reduce your level of cofidece. (That may ot be a useful solutio.) You eed to thik about your margi of error whe you desig your study. To get a arrower iterval without givig up cofidece, you eed to have less variability. You ca do this with a larger sample Slide 13 Key Poits: What Ca Go Wrog? (cot.) Choosig Your Sample Size: I geeral, the sample size eeded to produce a cofidece iterval with a give margi of error at a give cofidece level is: where z* is the critical value for your cofidece level. To be safe, roud up the sample size you obtai. 2 ˆp ˆq z ME 2 Violatios of Assumptios: Watch out for biased samples keep i mid what you leared i Chapter 2. Thik about idepedece. Slide 14 7
Check coditios: Radom: Gallup surveyed a radom sample of 439 U.S. tees. Normal: We check the couts of success ad failures pˆ 439*.56 246 ad (1 pˆ) 439*.44 193 The couts of successes ad failures are both 10 Idepedet: Sice Gallup sampled without replacemet, we eed to check the 10% coditio. At least 10(439) = 4390 U.S. tees aged 13 to 17. Fid the 95% z value: EXAMPLE: Tees Say Sex Ca Wait Cofidece iterval for p Fid: ˆp =246/439=.56 Calculate the cofidece iterval: p ˆ z * Calculator commad: ivnorm(.025,0,1) = -1.96 or ivnorm(.975,0,1) = 1.96 p ˆ (1 p ˆ ) (0.56)(0.44) 0.56 1.96 439 APPENDIX 0.56 0.046 (0.514,0.606) Coclude (i cotext): We are 95% cofidet that the iterval from.514 to.606 captures the true proportio of 13-to 17-year-olds i the Uited States who would say that tees should wait util marriage to have sex. Slide 15 EXAMPLE: Choosig Your Sample Size APPENDIX The Customer Service Problem Here is how to determie the sample size eeded to estimate p withi 0.03 with 95% cofidece. The critical value for 95% cofidece is z* = 1.96. Sice the compay presidet wats a margi of error of o more tha 0.03, we eed to solve the equatio 1.96 p ˆ (1 p ˆ ) 0.03 Multiply both sides by square root ad divide both sides by 0.03. Square both sides. Substitute 0.5 for the sample proportio to fid the largest ME possible. 1.96 0.03 p ˆ (1 p ˆ ) 2 1.96 p ˆ (1 p ˆ ) 0.03 2 1.96 (0.5)(1 0.5) 0.03 1067.111 ME z * ˆp ˆq We roud up to 1068 respodets to esure the margi of error is o more tha 0.03 at 95% cofidece. Slide 16 8