Confidence Intervals for the Population Proportion p

 Dale Carpenter
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1 Cofidece Itervals for the Populatio Proportio p The cocept of cofidece itervals for the populatio proportio p is the same as the oe for, the samplig distributio of the mea, x. The structure is idetical: estimate ± margi of error The assumptios eeded for cofidece itervals of proportios, is the same as for meas. However, it is prudet to poit out some very importat features that were metioed i 5.1 The distributio for is discrete. However, we will use a ormal approximatio, provided we ca meet the secod coditio that follows. I order to use the formula I will propose we must satisfy the coditio that p 10 ad (1 p) 10. If this coditio is ot met, we should ot use the formula below. Also all the assumptios of a biomial settig have to be met. estimate ± margi of error ± p(1 ˆ  p) ˆ Z Take a close look at the compoets of this formula: The mea for the distributio of sample proportios,, is p, µ. Ad the stadard deviatio is give by σ µ p, σ Sice we are usig zscores, Z, ad the associated probabilities of a ormal distributio, we the are assumig that the distributio of is such that the probability calculatios ca be approximated usig a ormal desity curve; thus the requiremet that we meet the coditio p 10 ad (1 p) 10. There is oe problem with the formula. You will otice that the purpose of the cofidece iterval is to estimate p. But the formula for σ eeds the use of p itself! How do we get aroud this quadary? We will use the estimate.
2 Calculatig the Sample Size The formula for calculatig the sample size for a give margi of error, m, ivolves solvig the equatio p(1 ˆ  p) ˆ m Z for. Z ˆ ˆ Whe we solve for we get, p(1  p). As you ca see we agai have a problem. We wat to m calculate the size of for a give value of m. The calculatio requires us to provide a value for, which we do ot have yet (we are just tryig to estimate how may observatios we eed to calculate ). What do we do? I am goig to replace with p. I will give you two choices for the value of p. Z m p(1  p) You must remember the purpose of why we create a cofidece iterval as explaied i 6.1. We wat to estimate a umber (the parameter) by creatig a iterval where our parameter may lie, ad (here is the crucial part) let everyoe kow how ofte, i the log ru, our iterval will cotai the desired parameter. Remember the problem from sectio 5.1, problem 5.7. The problem wated to illustrate that as the probability of success, p, got closer to 0 or 1, the stadard deviatio, goes to zero. Ad you ca see from the graph below that whe p 0.5 this results i the largest stadard deviatio possible for a fixed sample size. So below, for 15, if p 0.5 the σ is just about. σ Stadard Deviatio Stadard Deviatio i terms of p, probability of success for Probability of Success, p
3 So, if we make p 0.5 i the calculatio to fid the sample size, we would kow that whatever iterval we calculate, we ca guaratee the cofidece level. I eed to give you a width I ca guaratee with my cofidece level. The guaratee of course is how ofte my procedure will create a iterval cotaiig the parameter p, i this case. So you basically have two choices. Use the worst case sceario, p 0.5. Aother is to use p your best guess of p. How does this make sese? Now, suppose that I sell you a car, ad I make the guaratee that you should have o problems with the car for two years. Now I kow that i reality there is a good chace that othig will break dow with the car after four years of use, but I am ot as certai; I ca t guaratee it. So I give you a time period that I kow ca guaratee. Here is a Example. I wat to estimate the proportio of times a ball lads o 18 red i a roulette wheel. A roulette wheel had 18 red, 18 black ad two gree slots marked 0 ad 00; a total of 38 slots. I wat to calculate a 99% cofidece iterval, with a margi of error of at most 1%. What should be the sample size I gather? I kow that m 0.01 (for the 1%). Sice I eed a 99% cofidece iterval I kow Z.576. I this case I kow that the proportio I am lookig for is 18/38, so I could use p 18/38. Z m p(1  p) Thus Yes, I would eed to observe 16,544 plays i the hope to get my estimate withi 1% of the actual 99% of the time. What if I did ot kow that p 18/38? What would I have used for p? Oe optio is to use the value that gives the largest sample size for a give cofidece iterval; ,590. This results i a additioal 46 observatios i this case.
4 So we have two choices, either use the worst case sceario or the sample proportio as the estimate, for p i the calculatio of the stadard deviatio. Here is aother fu fact. Sice, we ow admit that the stadard deviatio, to the estimate as the stadard error, σ, is ot kow to us either, we refer SE.. p (1 p ) As was also metioed i 6.1 as the sample size goes up the margi of error decreases. The graph to the right shows this relatioship. Z The formula, p (1 p ) m, gives the sample size for a wated margi of error, m. Agai, you eed to estimate the value for p. I practice you would either use p 0.5 or you would use a best guess for p. Sometimes you ca use previous studies as a basis for that guess.
5 Homework Problems For Cofidece Itervals (p) 8.1 I each of the followig cases state whether or ot a ormal approximatio to the biomial should be used for a cofidece iterval for the populatio proportio p. a. 30 ad we estimate p will equal 0.9 b. 5 ad we estimate p will equal 0.5 c. 100 ad we estimate p will equal 0.04 d. 600 ad we estimate p will equal Whe tryig to hire maagers ad executives, compaies sometimes verify the academic credetials described by the applicats. Oe compay that performs these checks summarized their fidigs for a sixmoth period. Of the 84 applicats whose credetials were checked, 15 lied about havig a degree. a. Fid the proportio of applicats who lied about havig a degree ad the stadard error. b. Cosider these data to be radom sample of credetials from a large collectio of similar applicats. Give a 90% cofidece iterval for the true proportio of applicats who lie about havig a degree. 8.3 I recet years over 70% of firstyear college studets respodig to a atioal survey have idetified beig welloff fiacially as a importat persoal goal. A state uiversity fids that 13 of a SRS of 00 of its firstyear studets say that this goal is importat. Give 95% cofidece iterval for the proportio of all firstyear studets at the uiversity who would idetify beig welloff as a importat persoal goal. 8.4 The Gallup Poll asked a sample of 1785 U.S. adults, Did you, yourself, happe to atted church of syagogue i the last 7 days? Of the respodets, 750 said Yes. Suppose that the poll sample was a SRS. a. Give a 99% cofidece iterval for the proportio of all U.S. adults who atteded church or syagogue durig the week precedig the poll. b. Does the results provide good evidece that less tha half of the populatio atteded church or syagogue. c. How large a sample would be required to obtai a margi of error of ± 0.01 i a 99% cofidece iterval for the proportio who atted church or syagogue? (Use Gallup s result as the guessed value of p.)
6 Aswers 8.1 a No b. Yes, c. No. d. Yes. 8..a 15 84, S.E b (0.1098, 0.473) 8.3 (0.5943, 0.757) 8.4a. (0.3901, ) 8.4b. Yes, sice our iterval does ot iclude 0.5, half. 8.4c. 16,167.
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