# Overview. Learning Objectives. Point Estimate. Estimation. Estimating the Value of a Parameter Using Confidence Intervals

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1 Overview Estimatig the Value of a Parameter Usig Cofidece Itervals We apply the results about the sample mea the problem of estimatio Estimatio is the process of usig sample data estimate the value of a populatio parameter We will quatify the accuracy of our estimatio process Learig Objectives Compute a poit estimate of the populatio mea The Logic i Costructig Cofidece Itervals about a Populatio Mea whe Populatio Stadard Deviatio is Kow Costruct ad iterpret a cofidece iterval about the populatio mea (assumig the populatio stadard deviatio is kow) Uderstad the role of margi of error i costructig a cofidece iterval Determie the sample size ecessary for estimatig the populatio mea withi a specified margi of error Estimatio The eviromet of our problem is that we wat estimate the value of a ukow populatio mea The process that we use is called estimatio This is oe of the most commo goals of statistics Poit Estimate Estimatio ivolves two steps Step 1 obtai a specific umeric estimate, this is called the poit estimate Step quatify the accuracy ad precisio of the poit estimate The first step is relatively easy The secod step is why we eed statistics 1

2 Examples of Poit Estimate Some examples of poit estimates are The sample mea estimate the populatio mea The sample stadard deviatio estimate the populatio stadard deviatio The sample proportio estimate the populatio proportio The sample media estimate the populatio media Precisio of Poit Estimate The most obvious poit estimate for the populatio mea is the sample mea Now we will use the material o the samplig distributio of sample mea quatify the accuracy ad precisio of this poit estimate Example A example of what we wat quatify We wat estimate the miles per gallo for a certai car We test some umber of cars We calculate the sample mea it is 7 7 miles per gallo would be our best guess Example (cotiued) How sure are we that the gas ecoomy is 7 ad ot 8.1, or 5.? We would like make a statemet such as We thik that the mileage is 7 mpg ad we re pretty sure that we re ot o far off Iterval Estimatio A cofidece iterval for a ukow parameter is a iterval of umbers Compare this a poit estimate which is just oe umber, ot a iterval of umbers ( a rage of umbers) The level of cofidece represets the expected proportio of itervals that will cotai the parameter if a large umber of differet samples is obtaied The cofidece iterval quatifies the accuracy ad precisio of the poit estimate Iterpret Cofidece level What does the level of cofidece represet? If we obtai a series of 50 radom samples from a populatio of iterest Follow a process for calculatig cofidece itervals for populatio mea with a 90% level of cofidece from each of the sample meas The, we would expect that 90% of those 50 cofidece itervals (or about 45) would cotai our populatio mea

3 Cofidece Level The level of cofidece is always expressed as a percet The level of cofidece is described by a parameter α (i.e.,alpha) The level of cofidece is (1 α) 100% Whe α =.05, the (1 α) =.95, ad we have a 95% level of cofidece Whe α =.01, the (1 α) =.99, ad we have a 99% level of cofidece Cofidece Iterval If we expect that a method would create itervals that cotai the populatio mea 90% of the time, we call those itervals 90% cofidece itervals If we have a method for itervals that cotai the populatio mea 95% of the time, those are 95% cofidece itervals Ad so forth Summary To tie the defiitios gether We are usig the sample mea estimate the populatio mea..(poit estimate) With each specific sample, we ca costruct a,for istace, 95% cofidece iterval estimate the populatio mea (Iterval estimate) 95% cofidece iterval tells you that If we take samples repeatedly, we expect that 95% of these itervals would cotai the populatio mea Example Back our 7 miles per gallo car We thik that the mileage is 7 mpg ad we re pretty sure that we re ot o far off Puttig i umbers (quatify the accuracy) We estimate the gas mileage is 7 mpg ad we are 90% cofidet that the real mileage of this model of car is betwee 5 ad 9 miles per gallo Example (cotiued) We estimate the gas mileage is 7 mpg This is our poit estimate ad we are 90% cofidet that Our cofidece level is 90% (which is 1- α, i.e. α = 0.10) the real mileage of this model of car The populatio mea is betwee 5 ad 9 miles per gallo Our cofidece iterval is (5, 9) Kow Populatio Stadard Deviatio First, we assume that we kow the stadard deviatio of the populatio () This is ot very realistic but we eed it for right ow itroduce how costruct a cofidece iterval We ll solve this problem i a better way (where we do t kow what is) later but first we ll do this oe 3

5 Example For our car mileage example Assume that the sample mea was 7 mpg Assume that we tested a sample of 40 cars Assume that we kew that the populatio stadard deviatio was 6 mpg The our 95% cofidece iterval estimate for the true/populatio mea mileage would be 6 7 ± or 7 ± 1.9 Critical Value If we wated compute a 90% cofidece iterval, or a 99% cofidece iterval, etc., we would just eed fid the right stadard ormal value (istead of 1.96 for a 95% cofidece iterval) called critical value Frequetly used cofidece levels, ad their critical values, are 90% correspods % correspods % correspods.575 Critical Value The umbers 1.645, 1.960, ad.575 are writte as a form of Z α where α is the area the right of the Z value. z 0.05 = P(Z 1.645) =.05 [use TI Calcular: ivnorm(.95,0,1) = 1.645)] z 0.05 = P(Z 1.960) =.05 [ivnorm(0.975,0,1) = 1.960] z =.575 P(Z.575) =.005 [ivnorm(0,995,0.1) =.575] where Z is a stadard ormal radom variable How Determie Critical Value? Why do we use Z 0.05 for 95% cofidece? To be withi somethig 95% of the time We ca be o low.5% of the time We ca be o high.5% of the time Thus the 5% cofidece that we do t have is split as.5% beig o high ad.5% beig o low Critical Value z α/ for Cofidece Level 1 α I geeral, for a (1 α) 100% cofidece iterval, we eed fid z α/, the critical Z-value z α/ is the value such that P(Z z α/ ) = α/ Critical Value z α/ for 1 α Cofidece Level Oce we kow these critical values for the ormal distributio, the we ca costruct cofidece itervals for the populatio mea x z / α x + z / α 5

7 Sample Size Determiatio Determie the sample size ecessary for estimatig the populatio mea withi a specified margi of error Ofte we have the reverse problem where we wat a experimet achieve a particular accuracy of the estimatio. That is, we wat make sure the populatio mea ca be estimated withi a target margi of error from a sample mea. Sice the sample size will affect the margi of error, we wat fid the sample size () eeded achieve a particular size of margi of error i estimatio. Sample size determiatio is eeded i desigig a experimetal ivestigatio before the data collectio. Example For our car miles per gallo, we had = 6 If we wated our margi of error be 1 for a 95% cofidece iterval, the we would eed = = 1 Solvig for would get us = (1.96 6) or that = 138 cars would be eeded Sample Size Determiatio We ca write this as a formula The sample size eeded result i a margi of error E for (1 α) 100% cofidece is z = α / E Usually we do t get a iteger for, so we would eed take the ext higher umber (the oe lower would t be large eough) Summary We ca costruct a cofidece iterval aroud a poit estimar if we kow the populatio stadard deviatio The margi of error is calculated usig, the sample size, ad the appropriate Z- value We ca also calculate the sample size eeded obtai a target margi of error Cofidece Itervals about a Populatio Mea i Practice where the Populatio Stadard Deviatio is Ukow 7

9 Differece betwee Z ad t So what s differet? Ulike the ormal, there are may differet stadard t-distributios There is a stadard oe with 1 degree of freedom There is a stadard oe with degrees of freedom There is a stadard oe with 3 degrees of freedom Etc. The umber of degrees of freedom is crucial for the t-distributios t-statistic Whe is kow, the z-score x µ z = / follows a stadard ormal distributio Whe is ot kow, the t-statistic x µ t = s / follows a t-distributio with 1(sample size mius 1) degrees of freedom t-distributio Comparig three curves The stadard ormal curve The t curve with 14 degrees of freedom The t curve with 4 degrees of freedom Determie t-values Calculatio of t-distributio The calculatio of t-distributio values t α ca be doe i similar ways as the calculatio of ormal values z α Usig tables Usig techology TI graphig Calcular Use a t-table show fid a critical value Upper critical values of Studet's t distributio with ν degrees of freedom Or use TI graphig calcular fid a critical value: for istace, t 0.05 & df = 3 = ivt(0.95,3) =.3534 t 0,01& df = 11 = ivt(0.99,11) =.7187 Probability of exceedig the critical value

10 Critical values t Critical values for various degrees of freedom for the t- distributio are (compared the ormal) Normal Degrees of Freedom Ifiite t Costruct ad iterpret a t-cofidece iterval about a populatio mea Note: Whe the sample size is large, a t distributio is close a z distributio z-score ad t-score The differece betwee the two formulas x µ z = / x µ t = s/ is that the sample stadard deviatio s is used approximate the populatio stadard deviatio The z-score has a ormal distributio, the t-statistic (or the t-score) has a t-distributio 95% Cofidece iterval for mea with ukow A 95% cofidece iterval, with ukow, is x t s where t 0.05 is the critical value for the t-distributio with ( 1) degrees of freedom x + t0. 05 s Note: Compare it the 95% cofidece iterval, with a kow : x z x + z Critical Value t α/ correspodig Cofidece Level 1 α The differet 95% cofidece itervals with t 0.05 would be For = 6, the sample mea ±.571 s / 6 For = 16, the sample mea ±.131 s / 16 For = 31, the sample mea ±.04 s / 31 For = 101, the sample mea ± s / 101 For = 1001, the sample mea ± 1.96 s / 1001 Whe is kow, the sample mea ± / Cofidece iterval for mea with ukow I geeral, the (1 α) 100% cofidece iterval, whe is ukow, is x tα / s x + tα / s where t α/ is the critical value for the t-distributio with ( 1) degrees of freedom 10

12 Learig Objectives Cofidece Itervals about a Populatio Proportio Obtai a poit estimate for the populatio proportio Costruct ad iterpret a cofidece iterval for the populatio proportio Determie the sample size ecessary for estimatig a populatio proportio withi a specified margi of error Mea & Proportio Obtai a poit estimate for the populatio proportio So far, we leared calculate cofidece itervals for the populatio mea, whe we kew ad We also leared calculate cofidece itervals for the mea, whe we did ot kow Here, we ll lear how costruct cofidece itervals for situatios whe we are aalyzig a populatio proportio The issues ad methods are quite similar Sample Proportio Whe we aalyze the populatio mea, we use the sample mea as the poit estimate The sample mea is our best guess for the populatio mea Whe we aalyze the populatio proportio, we use the sample proportio as the poit estimate The sample proportio is our best guess for the populatio proportio Proportio Poit Estimate Usig the sample proportio is the atural choice for the poit estimate If we are doig a poll, ad 68% of the respodets said yes our questio, the we would estimate that 68% of the populatio would say yes our questio also The sample proportio is writte as pˆ 1

13 Costruct ad iterpret a cofidece iterval for the populatio proportio Cofidece Iterval for Mea versus Proportio Cofidece itervals for the populatio mea are Cetered at the sample mea Plus ad mius z α/ times the stadard deviatio of the sample mea (the stadard error from the samplig distributio) Similarly, cofidece itervals for the populatio proportio will be Cetered at the sample proportio Plus ad mius z α/ times the stadard deviatio of the sample proportio Samplig Distributio of Proportio We have already studied the distributio of the sample proportio is approximately ormal with pˆ = µ p ˆ = p p( 1 p) uder most coditios We use this costruct cofidece itervals for the populatio proportio Cofidece Iterval for Populatio Proportio The (1 α) 100% cofidece iterval for the populatio proportio is from pˆ zα / where z α/ is the critical value for the ormal distributio Note: That is, pˆ( 1 pˆ) pˆ + zα / pˆ( 1 pˆ) sample proportio ± z α/ stadard error of sample proportio Margi of Error Like for cofidece itervals for populatio meas, the quatity zα / pˆ( 1 pˆ) is called the margi of error Example We polled = 500 voters (This a sample of voters) Whe asked about a ballot questio, pˆ = 47% of them were i favor Obtai a 99% cofidece iterval for the populatio proportio i favor of this ballot questio (α = 0.005) 13

14 Example (cotiued) The critical value z =.575, so = = Determie the sample size ecessary for estimatig a populatio proportio withi a specified margi of error or (0.41, 0.53) is a 99% cofidece iterval for the populatio proportio Sample Size Determiatio We ofte wat kow the miimum sample size obtai a target margi of error for estimatig the populatio proportio A commo use of this calculatio is i pollig how may people eed be polled for the result have a certai margi of error News sries ofte say the latest polls show that so-ad-so will receive X% of the votes with a E% margi of error Example 1 For our pollig example, how may people eed be polled so that we are withi 1 percetage poit with 99% cofidece? The margi of error is pˆ (1 pˆ ) z α / which must be 0.01 We have a problem, though what is pˆ? Two choices of pˆ Example 1 (cotiued) If we try figure out the sample size i the experimetal desig stage before collectig data, the we do ot have sample data calculate pˆ. A way aroud this is that usig pˆ = 0.5 will always yield a sample size that is large eough. We ca also use a estimates pˆ from a previous study (hisric data) calculate the sample size. I our case, if we usig so ad = 16,577 pˆ = = = , the we have 14

15 Example 1 (cotiued) We uderstad ow why political polls ofte have a 3 or 4 percetage poits margi of error Sice it takes a large sample ( = 16,577) get be 99% cofidet withi 1 percetage poit, the 3 or 4 percetage poits margi of error targets are good compromises betwee accuracy ad cost effectiveess Sample Size Determiatio We ca write this as a formula The sample size eeded result i a margi of error E% for (1 α) 100% cofidece for a populatio proportio is ( Z ) pˆ ( 1 pˆ ) α / = ( E% ) Usually we do t get a iteger for, so we would eed take the ext higher umber (the oe lower would t be large eough) Example Determie the sample size ecessary estimate the true proportio of laborary mice with a certai geetic defect. We would like the estimate be withi with 95% cofidece. Solutio: 1. Level of cofidece: 1 α = 0.95, z α/ = Desired maximum error is E = No estimate of p give, use pˆ = Use the formula for : ( Z ) pˆ ( 1 pˆ ) α = ( 196. ) ( 05. )( 05. ) ( E% ) ( ) / = = Example (cotiued) Suppose we kow the geetic defect occurs i approximately 1 of 80 aimals Use: pˆ = 1 / 80 = = ( Z ) pˆ ( 1 pˆ ) / α ( E% ) ( 196. ) ( )( ) ( ) = = Note: As illustrated here, it is a advatage have some idicatio of the value expected for p, especially as p becomes icreasigly further from 0.5 Summary We ca costruct cofidece itervals for populatio proportios i much the same way as for populatio meas We eed use the formula for the stadard deviatio of the sample proportio We ca also compute the miimum sample size eeded for a desired level of accuracy Which Procedure Do I Use? 15

16 Overview There are three differet cofidece iterval calculatios covered i this uit It ca be cofusig which oe is appropriate for which situatio I should use the ormal o, the t o the??? Which Parameter? The oe mai questio right at the begiig Which parameter are we tryig estimate? A mea? A proportio? This the sigle most importat questio z-iterval or t-iterval? I aalyzig populatio meas Is the populatio variace kow? If so, the we ca use the ormal distributio If the populatio variace is ot kow If we have eough data (30 or more values), we still ca use the ormal distributio If we do t have eough data (9 or fewer values), we should use the Studet's t-distributio We do t have ask this questio i the aalysis of proportios z-iterval for mea For the aalysis of a populatio mea If The data is OK (reasoably ormal) The variace is kow the we ca use the ormal distributio with a cofidece iterval of x zα / x + zα / t-iterval for mea For the aalysis of a populatio mea If The data is OK (reasoably ormal) The variace is NOT kow the we ca use the Studet's t-distributio with a cofidece iterval of x tα / s x + tα / s z-iterval for Proportio For the aalysis of a populatio proportio If sample size is large eough, the we ca use the proportios method with a cofidece iterval of pˆ zα / pˆ( 1 pˆ) pˆ + zα / pˆ( 1 pˆ) 16

17 Summary The mai questios that determie the cofidece iterval use: Is it a Populatio mea? Populatio proportio? I the case of a populatio mea, we eed determie Is the populatio variace kow? Does the data look reasoably ormal? Estimatig the Value of a Parameter Usig Cofidece Itervals Summary We ca use a sample {mea, proportio} estimate the populatio {mea, proportio} I each case, we ca use the appropriate samplig distributio of the sample statistic costruct a cofidece iterval aroud our estimate The cofidece iterval expresses the cofidece we have that our calculated iterval cotais the true parameter 17

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