Unit 5: Estimating with Confidence

Transcription

1 Uit 5: Estimatig with Cofidece Sectio 8.2 The Practice of Statistics, 4 th editio For AP* STARNES, YATES, MOORE

2 Uit 5 Estimatig with Cofidece Cofidece Itervals: The Basics Estimatig a Populatio Mea

3 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

4 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. 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.

5 Data Collectio Shows 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 How ca we use this iformatio to fid a cofidece iterval for 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.

6 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 Normal: Both p ad (1 p) must be greater tha 10. Sice we do t kow p, we check that pˆ ad (1 pˆ) 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.

7 Costructig a Cofidece Iterval for p We ca use the geeral formula from Sectio 10.1 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.

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 * 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.

9 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.

10 Oe-Sample z Iterval for a Populatio Proportio State: We wat to calculate ad iterpret a 90% cofidece iterval for the proportio of red beads i the cotaier. Do: Pla: z sample proportio = 107/251 = We checked the coditios earlier For a 90% cofidece level, z* = statistic ± (critical value) (stadard deviatio of the statistic) ˆ p z * Coclude: p ˆ (1 p ˆ ) (0.426)( ) (0.375, 0.477) We are 90% cofidet that the iterval from to captures the actual proportio of red beads i the cotaier.

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 * ˆ p (1 ˆ p ) 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.

12 Example: Customer Satisfactio 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?

13 Example: Customer Satisfactio Determie the sample size eeded to estimate p withi 0.03 with 95% cofidece. The critical value for 95% cofidece is z* = 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 Square both sides. Substitute 0.5 for the sample proportio to fid the largest ME possible. p 1.96 ˆ (1 p ˆ ) ˆ p (1 ˆ p ) p ˆ (1 p ˆ ) (0.5)(1 0.5) We roud up to 1068 respodets to esure the margi of error is o more tha 0.03 at 95% cofidece.

14 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

15 Homework Textbook Chapter 12 # s, 1-4, 6, 7, 10, 11 Additioal otes packet Read ad aswer all questios, pgs