CHAPTER 8 Estimatig with Cofidece 8.2 Estimatig a Populatio Proportio The Practice of Statistics, 5th Editio Stares, Tabor, Yates, Moore Bedford Freema Worth Publishers
Estimatig a Populatio Proportio Learig Objectives After this sectio, you should be able to: STATE ad CHECK the Radom, 10%, ad Large Couts coditios for costructig a cofidece iterval for a populatio proportio. DETERMINE critical values for calculatig a C % cofidece iterval for a populatio proportio usig a table or techology. CONSTRUCT ad INTERPRET a cofidece iterval for a populatio proportio. DETERMINE the sample size required to obtai a C % cofidece iterval for a populatio proportio with a specified margi of error. The Practice of Statistics, 5 th Editio 2
Activity: The Beads Your teacher has a cotaier full of differet colored beads. Your goal is to estimate the actual proportio of red beads i the cotaier. Form teams of 3 or 4 studets. Determie how to use a cup to get a simple radom sample of beads from the cotaier. Each team is to collect oe SRS of beads. Determie a poit estimate for the ukow populatio proportio. Fid a 95% cofidece iterval for the parameter p. Cosider ay coditios that are required for the methods you use. Compare your results with the other teams i the class. The Practice of Statistics, 5 th Editio 3
Coditios for Estimatig p Suppose oe SRS of beads resulted i 107 red beads ad 144 beads of aother color. The poit estimate for the ukow proportio p of red beads i the populatio would be p ˆ = 107 251 = 0.426 How ca we use this iformatio to fid a cofidece iterval for p? If the sample size is large eough that both p ad (1- p) are at least 10, the samplig distributio of p ˆ is approximately Normal. The mea of the samplig distributio of ˆ p is p. The stadard deviatio of the samplig distributio of p ˆ is s p ˆ = p(1- p). I practice, we do ot kow the value of p. If we did, we would ot eed to costruct a cofidece iterval for it! I large samples, p ˆ will be close to p, so we will replace p with p ˆ i checkig the Normal coditio. The Practice of Statistics, 5 th Editio 4
Coditios for Estimatig p Before costructig a cofidece iterval for p, you should check some importat coditios Coditios for Costructig a Cofidece Iterval About a Proportio Radom: The data come from a well-desiged radom sample or radomized experimet. o 10%: Whe samplig without replacemet, check that 1 10 N ˆp ad (1- ˆp) Large Couts: Both are at least 10. The Practice of Statistics, 5 th Editio 5
Costructig a Cofidece Iterval for p We ca use the geeral formula from Sectio 8.1 to costruct a cofidece iterval for a ukow populatio proportio p: statistic ± (critical value) (stadard deviatio of statistic) The sample proportio p ˆ is the statistic we use to estimate p. Whe the Idepedet coditio is met, the stadard deviatio of the samplig distibutio of p ˆ is s p ˆ = p(1- p) Sice we do't kow p, we replace it with the sample proportio p ˆ. This gives us the stadard error (SE) of the sample proportio: p ˆ (1- p ˆ ) Whe the stadard deviatio of a statistic is estimated from data, the results is called the stadard error of the statistic. The Practice of Statistics, 5 th Editio 6
Fidig a Critical Value How do we fid the critical value for our cofidece iterval? statistic ± (critical value) (stadard deviatio of statistic) If the Large Couts coditio is met, we ca use a Normal curve. To fid a level C cofidece iterval, we eed to catch the cetral area C uder the stadard Normal curve. To fid a 95% cofidece iterval, we use a critical value of 2 based o the 68-95-99.7 rule. Usig Table A or a calculator, we ca get a more accurate critical value. Note, the critical value z* is actually 1.96 for a 95% cofidece level. The Practice of Statistics, 5 th Editio 7
Example: Fidig a Critical Value Use Table A to fid the critical value z* for a 80% cofidece iterval. Assume that the Large Couts coditio is met. The closest etry is z = 1.28. Sice we wat to capture the cetral 80% of the stadard Normal distributio, we leave out 20%, or 10% i each tail. Search Table A to fid the poit z* with area 0.1 to its left. z.07.08.09 1.3.0853.0838.0823 1.2.1020.1003.0985 1.1.1210.1190.1170 So, the critical value z* for a 80% cofidece iterval is z* = 1.28. The Practice of Statistics, 5 th Editio 8
Oe-Sample z Iterval for a Populatio Proportio Oce we fid the critical value z*, our cofidece iterval for the populatio proportio p is statistic ± (critical value) (stadard deviatio of statistic) = ˆ p ± z * p ˆ (1- p ˆ ) Oe-Sample z Iterval for a Populatio Proportio Whe the coditios are met, a C% cofidece iterval for the ukow proportio p is ˆ p ± z * where z* is the critical value for the stadard Normal curve with C% of its area betwee z* ad z*. ˆ p (1- ˆ p ) The Practice of Statistics, 5 th Editio 9
Oe-Sample z Iterval for a Populatio Proportio Suppose you took a SRS of beads from the cotaier ad got 107 red beads ad 144 white beads. Calculate ad iterpret a 90% cofidece iterval for the proportio of red beads i the cotaier. Your teacher claims 50% of the beads are red. Use your iterval to commet o this claim. z.03.04.05 1.7.0418.0409.0401 1.6.0516.0505.0495 1.5.0630.0618.0606 ˆ p ± z * p ˆ (1- p ˆ ) Sample proportio = 107/251 = 0.426 We checked the coditios earlier. For a 90% cofidece level, z* = 1.645 We are 90% cofidet that the iterval from 0.375 to 0.477 captures the true proportio of red beads i the cotaier. = 0.426 ±1.645 = 0.426 ± 0.051 = (0.375, 0.477) (0.426)(1-0.426) 251 Sice this iterval gives a rage of plausible values for p ad sice 0.5 is ot cotaied i the iterval, we have reaso to doubt the claim. The Practice of Statistics, 5 th Editio 10
The Four Step Process We ca use the familiar four-step process wheever a problem asks us to costruct ad iterpret a cofidece iterval. Cofidece Itervals: A Four-Step Process State: What parameter do you wat to estimate, ad at what cofidece level? Pla: Idetify the appropriate iferece method. Check coditios. Do: If the coditios are met, perform calculatios. Coclude: Iterpret your iterval i the cotext of the problem. The Practice of Statistics, 5 th Editio 11
Choosig the Sample Size I plaig a study, we may wat to choose a sample size that allows us to estimate a populatio proportio withi a give margi of error. The margi of error (ME) i the cofidece iterval for p is ME = z * p ˆ (1- p ˆ ) z* is the stadard Normal critical value for the level of cofidece we wat. Because the margi of error ivolves the sample proportio ˆ p, we have to guess the latter value whe choosig. There are two ways to do this: Use a guess for ˆ p based o past experiece or a pilot study Use ˆ p = 0.5 as the guess. ME is largest whe ˆ p = 0.5 The Practice of Statistics, 5 th Editio 12
Choosig the Sample Size I plaig a study, we may wat to choose a sample size that allows us to estimate a populatio proportio withi a give margi of error. Calculatig a Cofidece Iterval To determie the sample size that will yield a level C cofidece iterval for a populatio proportio p with a maximum margi of error ME, solve the followig iequality for : p ˆ (1- p ˆ ) z * ME where p ˆ is a guessed value for the sample proportio. The margi of error will always be less tha or equal to ME if you take the guess p ˆ to be 0.5. The Practice of Statistics, 5 th Editio 13
Example: Determiig 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 she will be required to pay for. Oe critical questio is the degree of satisfactio with the compay s customer service, measured o a five-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 two highest levels o the five-poit scale). She decides that she wats the estimate to be withi 3% (0.03) at a 95% cofidece level. How large a sample is eeded? The Practice of Statistics, 5 th Editio 14
Example: Determiig sample size Problem: Determie the sample size eeded to estimate p withi 0.03 with 95% cofidece. The critical value for 95% cofidece is z* = 1.96. We have o idea about the true proportio p of satisfied customers, so we decide to use p-hat = 0.5 as our guess. Because the compay presidet wats a margi of error of o more tha 0.03, we eed to solve the equatio: margi of error < 0.03 The Practice of Statistics, 5 th Editio 15
Example: Determiig sample size Because the compay presidet wats a margi of error of o more tha 0.03, we eed to solve the equatio Multiply both sides by square root ad divide both sides by 0.03. Square both sides. 1.96 ˆp(1- ˆp) 1.96 0.03 0.03 0.5(1-0.5) 2 æ 1.96 ö ç (0.5)(1-0.5) è 0.03ø 1067.111 We roud up to 1068 respodets to esure that the margi of error is o more tha 3%. The Practice of Statistics, 5 th Editio 16
Estimatig a Populatio Proportio Sectio Summary I this sectio, we leared how to STATE ad CHECK the Radom, 10%, ad Large Couts coditios for costructig a cofidece iterval for a populatio proportio. DETERMINE critical values for calculatig a C % cofidece iterval for a populatio proportio usig a table or techology. CONSTRUCT ad INTERPRET a cofidece iterval for a populatio proportio. DETERMINE the sample size required to obtai a C % cofidece iterval for a populatio proportio with a specified margi of error. The Practice of Statistics, 5 th Editio 17