Chapter 8: Estimatig with Cofidece Sectio 8.2 The Practice of Statistics, 4 th editio For AP* STARNES, YATES, MOORE
Chapter 8 Estimatig with Cofidece 8.1 Cofidece Itervals: The Basics 8.2 8.3 Estimatig a Populatio Mea
Sectio 8.2 Learig Objectives After this sectio, you should be able to CONSTRUCT ad INTERPRET a cofidece iterval for a populatio proportio DETERMINE the sample size required to obtai a level C cofidece iterval for a populatio proportio with a specified margi of error DESCRIBE how the margi of error of a cofidece iterval chages with the sample size ad the level of cofidece C
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 90% 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.
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 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,ˆ p will be close to p, so we will replace p with p ˆ i checkig the Normal coditio.
Coditios for Estimatig p Check the coditios for estimatig p from our sample. Radom: The class took a SRS of 251 beads from the cotaier. p ˆ 107 251 0.426 Normal: Both p ad (1 p) must be greater tha 10. Sice we do t kow p, we check that ˆ p 251 107 107 ad (1 p ˆ ) 251 1 107 144 251 251 The couts of successes (red beads) ad failures (o-red) are both 10. Idepedet: Sice the class sampled without replacemet, they eed to check the 10% coditio. At least 10(251) = 2510 beads eed to be i the populatio. The teacher reveals there are 3000 beads i the cotaier, so the coditio is satisfied. Sice all three coditios are met, it is safe to costruct a cofidece iterval.
Alterate Example The peies Ms. Smith s class wats to costruct a cofidece iterval for the proportio p of peies more tha 10 years old i their collectio. Their sample had 57 peies more tha 10 years old ad 45 peies that were at most 10 years old. Problem: Check that the coditios for costructig a cofidece iterval for p are met. Solutio: Radom: The class took a SRS of 102 peies from the collectio. Normal : pˆ 102 57 102 57 ad (1 p) 102 45 102 Both the umber of successes ad the umber of failures are at least 10. 45 Idepedet: Sice we are samplig without replacemet, the umber of peies i the populatio must be at least 10(102) = 1020. Sice there are more tha 2000 peies i Ms. Smith s collectio, the 10% coditio is met.
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 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 ) Defiitio: Whe the stadard deviatio of a statistic is estimated from data, the results is called the stadard error of the statistic.
Fidig a Critical Value How do we fid the critical value for our cofidece iterval? statistic (critical value) (stadard deviatio of statistic) If the Normal 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. For example, 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.
Fidig a Critical Value Use Table A to fid the critical value z* for a 80% cofidece iterval. Assume that the Normal 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.
Alterate Example 96% cofidece Problem: Use Table A to fid the critical value z* for a 96% cofidece iterval. Assume that the Normal coditio is met. Solutio: For a 96% cofidece iterval, we eed to capture the middle 96% of the stadard Normal distributio. This leaves out 2% i each tail. So, we wat to fid the z-score with a area of 0.02 to its left. The closest etry is z = 2.05, so the critical value we wat is z* = 2.05.
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 * Oe-Sample z Iterval for a Populatio Proportio Choose a SRS of size from a large populatio that cotais a ukow proportio p of successes. A approximate level C cofidece iterval for p is ˆ p z * p ˆ (1 p ˆ ) p ˆ (1 p ˆ ) where z* is the critical value for the stadard Normal curve with area C betwee z* ad z*. Use this iterval oly whe the umbers of successes ad failures i the sample are both at least 10 ad the populatio is at least 10 times as large as the sample.
Oe-Sample z Iterval for a Populatio Proportio 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 sample proportio = 107/251 = 0.426 1.7.0418.0409.0401 We checked the coditios earlier. 1.6.0516.0505.0495 For a 90% cofidece level, z* = 1.645 1.5.0630.0618.0606 statistic ± (critical value) (stadard deviatio of the statistic) ˆ p z * p ˆ (1 p ˆ ) 0.426 1.645 0.426 0.051 (0.375, 0.477) (0.426)(1 0.426) 251 We are 90% cofidet that the iterval from 0.375 to 0.477 captures the actual proportio of red beads i the cotaier. 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.
Alterate Example The peies Problem: Ms. Smith s class took a SRS of 102 peies ad discovered that 57 of the peies were more tha 10 years old. (a) Calculate ad iterpret a 99% cofidece iterval for p = the true proportio of peies from the collectio that are more tha 10 years old. The proportio of peies more tha 10 years old i the sample was = 57/102 = 0.559. The critical value for a 99% cofidece iterval ca be foud by lookig for the poit that has a area of 0.005 to the left. The calculator s ivnorm(0.005,0,1) gives 2.576 so the appropriate critical value for 99% cofidece is z* = 2.576. The 99% cofidece iterval is: pˆ z * pˆ(1 pˆ) 0.559 2.576 0.559(1 0.559 102 0.559 0.127 (0.432,0.686) We are 99% cofidet that the iterval from 0.432 to 0.686 captures the actual proportio of peies i the collectio that are more tha 10 years old. (b) Is it plausible that exactly 60% of all the peies i the collectio are more tha 10 years old? Explai. Yes, sice 0.6 is icluded i the cofidece iterval, it is plausible that 60% of all the peies i the collectio are more tha 10 years old.
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.
Alterate Example: Kissig the right way? Accordig to a article i the Sa Gabriel Valley Tribue (2-13-03), Most people are kissig the right way. That is, accordig to the study, the majority of couples tilt their heads to the right whe kissig. I the study, a researcher observed a radom sample 124 couples kissig i various public places ad foud that 83/124 (66.9%) of the couples tilted to the right. Costruct ad iterpret a 95% cofidece iterval for the proportio of all couples who tilt their heads to the right whe kissig. State: We wat to estimate p = the true proportio of couples that tilt their heads to the right whe kissig at the 95% cofidece level. Pla: We will use a oe-sample z iterval for p if the followig coditios are satisfied. Radom: The researcher observed a radom sample of couples. Normal : pˆ 83 10 ad (1 p) 4110 Idepedet: The umber of couples i the populatio is more tha 10(124) = 1240. pˆ(1 pˆ) 0.669(1 0.669) Do: p z * 0.669 1.96 0.669 0.083 (0.586,0.752) 124 Coclude: We are 95% cofidet that the iterval from 0.586 to 0.752 captures the true proportio of couples that tilt their heads to the right whe kissig.
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 p ˆ, we have to guess the latter value whe choosig. There are two ways to do this: Use a guess for p ˆ based o past experiece or a pilot study Use p ˆ 0.5 as the guess. ME is largest whe p ˆ 0.5 Sample Size for Desired Margi of Error 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.
Example: Customer Satisfactio Read the example o page 493. 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 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. p 1.96 ˆ (1 p ˆ ) 1.96 0.03 ˆ p (1 ˆ p ) 0.03 2 1.96 p ˆ (1 p ˆ ) 0.03 2 1.96 (0.5)(1 0.5) 0.03 1067.111 We roud up to 1068 respodets to esure the margi of error is o more tha 0.03 at 95% cofidece.
Alterate Example: Tattoos Suppose that you wated to estimate the p = the true proportio of studets at your school that have a tattoo with 95% cofidece ad a margi of error of o more tha 0.10. Problem: Determie how may studets should be surveyed to estimate p withi 0.10 with 95% cofidece. Solutio: Sice we do t have ay previous kowledge of the proportio of studets with a tattoo, we will use = 0.5 to estimate the sample size eeded. 1.96 0.5(1 0.5) 0.10 1.96 0.10 2 (0.5)(1 0.5) 96.04 So, we eed to survey at least 97 studets to estimate the true proportio of studets with a tattoo with 95% cofidece ad a margi of error of at most 0.10.
Sectio 8.2 Summary I this sectio, we leared that Cofidece itervals for a populatio proportio p are based o the samplig distributio of the sample proportio p ˆ. Whe is large eough that both p ad (1 p) are at least 10, the samplig distributio of p is approximately Normal. I practice, we use the sample proportio p ˆ to estimate the ukow parameter p. We therefore replace the stadard deviatio of p ˆ with its stadard error whe costructig a cofidece iterval. p ˆ (1 p ˆ ) The level C cofidece iterval for p is : p ˆ z *
Sectio 8.2 Summary I this sectio, we leared that Whe costructig a cofidece iterval, follow the familiar four-step process: STATE: What parameter do you wat to estimate, ad at what cofidece level? PLAN: Idetify the appropriate iferece method. Check coditios. DO: If the coditios are met, perform calculatios. CONCLUDE: Iterpret your iterval i the cotext of the problem. The sample size eeded to obtai a cofidece iterval with approximate margi of error ME for a populatio proportio ivolves solvig p ˆ (1 p ˆ ) z * ME for, where p ˆ is a guessed value for the sample proportio,ad z * is the critical value for the level of cofidece you wat. If you use p ˆ 0.5 i this formula, the margi of error of the iterval will be less tha or equal to ME.
Lookig Ahead I the ext Sectio We ll lear how to estimate a populatio mea. We ll lear about The oe-sample z iterval for a populatio mea whe σ is kow The t distributios whe σ is ukow Costructig a cofidece iterval for µ Usig t procedures wisely