Chapter 8: Estimatio of Mea & Proportio 8.1 Estimatio, Poit Estimate, ad Iterval Estimate 8.2 Estimatio of a Populatio Mea: σ Kow 8.3 Estimatio of a Populatio Mea: σ Not Kow 8.4 Estimatio of a Populatio Proportio: Large Samples 8-1 Itroductio Statistics is the sciece of learig from data Collectig data Summarizig ad aalyzig data Makig iferece & drawig coclusio, Iterpretatio Theoretical foudatio probability theory Statistics has two big topics i this class Summary statistics & Iferetial statistics Iferetial statistics Estimatio of populatio parameters, icludig poit ad iterval estimate Hypothesis testig of populatio parameters 8-2 1
8.1 Estimatio, Poit Estimate, ad Iterval Estimate What is estimatio of a parameter? The assigmet of value(s) to the parameter based o a value of the correspodig sample statistic is called estimatio. Estimator uobserved sample statistic, is a radom variable Estimate observed sample statistic, a umber. The estimatio procedure ivolves the followig steps Select a sample. Collect the required iformatio from the members of the sample. Calculate the value of the sample statistic. Assig value(s) to the correspodig populatio parameter Examples Average life time of certai brad of smart phoe after oe charge? Radom sample: 20 this type smart phoe What is estimator? What is estimate? 8-3 Poit & Iterval Estimates Poit estimate The value of a sample statistic that is used to estimate a parameter is called a poit estimate Why it is called poit estimate? How to obtai the poit estimate? Iterval estimate Istead of oe sigle value, a iterval (two eds), which is costructed aroud the poit estimate, is used It is also stated that how likely (95%, say) the iterval cotais the correspodig parameter How to costruct such a iterval estimate? 2 values 1 probability Why do we eed two estimates? 8-4 2
Example of Estimatio Istead of a poit estimate, we provide iterval estimate, i.e., poit estimate ± a umber. How to fid the a umber ofte referred as margi of error 8-5 Cofidece Level ad Cofidece Iterval Each iterval is costructed with regard to a give cofidece level ad is called a cofidece iterval. The cofidece iterval is give as Poit estimate ± Margi of error The cofidece level associated with a cofidece iterval states how much cofidece we have that this iterval cotais the true populatio parameter. The cofidece level is deoted by (1 α)100%. 8-6 3
8.2 Estimatio of a Populatio Mea: Kow Three possible cases with kow σ Case I: the populatio from which the sample is selected is ormally distributed. I this case, sample size is ot matter. Case II: the populatio is ot ormal, but the sample size is large (i.e., 30) Case III: the populatio is ot ormal AND the sample size is small (i.e., < 30) 8-7 Steps to Costruct Cofidece Iterval o μ Fudametal idea Two ed poits ad a probability (cofidece) Probability comes from a kow distributio Sample mea s distributio 2 X X ~ N (, ) ~ N (0,1) / Fid z such that P( z Z z) 0.95 (or i geeral 1 ) X Solve for from z z / Steps to obtai cofidece iterval o μ Poit estimate ----------------- X Critical value ------------------- z from stadard ormal distributio Margi of error E z Cofidece iterval ----------- ± E X 8-8 4
Fidig z for a 100(1 α)% Cofidece Level How to fid z? Left tail of z = 1 α/2 Table, Excel (ormsiv), or calculator (ivnorm) 8-9 Example 8.1 Normal Populatio A publishig compay has just published a ew college textbook. Before the compay decides the price at which to sell this textbook, it wats to kow the average price of all such textbooks i the market. The research departmet at the compay took a sample of 25 comparable textbooks ad collected iformatio o their prices. This iformatio produced a mea price of $145 for this sample. It is kow that the stadard deviatio of the prices of all such textbooks is $35 ad the populatio of such prices is ormal. a). What is the poit estimate of the mea price of all such college textbooks? b). Costruct a 90% cofidece iterval for the mea price of all such college textbooks. Solutio: a). Poit estimate of mea = sample mea x = $145 b). 90% C.I. for mea : Poit estimate x = $145 90% => z, i.e., P(Z < z) = 0.95. z = 1.65 E = z = 1.6535/5 = 1.657 = 11.55 90% C.I. for is 145 11.55 8-10 5
Example 8.2 Large Sample Accordig to Moebs Services Ic., a idividual checkig accout at major U.S. baks costs the baks more tha $380 per year. A recet radom sample of 600 such checkig accouts produced a mea aual cost of $500 to major U.S. baks. Assume that the stadard deviatio of aual costs to major U.S. baks of all such checkig accouts is $40. Make a 99% cofidece iterval for the curret mea aual cost to major U.S. baks of all such checkig accouts. Solutio: 99% C.I. for mea : 1. Poit estimate: 500 2. 99% => z, i.e., P(Z < z) = 0.995. z = 2.58 3. E = z = 2.581.633 = 4.21 4. 99% C.I. for is 500 4.21 8-11 Sample Size for the Estimatio of Mea Width of a Cofidece Iterval The width of a cofidece iterval depeds o the size of the margi of error, E z. Hece, the width of a cofidece iterval ca be cotrolled usig The value of z, which depeds o the cofidece iterval The sample size, Determiig the Sample Size for the Estimatio of μ Give the cofidece level ad the stadard deviatio of the populatio, the sample size that will produce a predetermied margi of error E of the cofidece iterval estimate of μ is 2 2 z 2 E 8-12 6
Example 8-3 A alumi associatio wats to estimate the mea debt of this year s college graduates. It is kow that the populatio stadard deviatio of the debts of this year s college graduates is $11,800. How large a sample should be selected so that the estimate with a 99% cofidece level is withi $800 of the populatio mea? Solutio 99% => z = 2.58 E = 800 Stadard deviatio σ = 11800 Therefore, = z 2 σ 2 / E 2 = 1448.18 1449 8-13 8.3 Estimatio of a Populatio Mea: σ ukow Three possible cases with ukow σ Case I: the populatio from which the sample is selected is ormally distributed. I this case, sample size is ot matter. Case II: the populatio is ot ormal, but the sample size is large (i.e., 30) Case III: the populatio is ot ormal AND the sample size is small (i.e., < 30)???? 8-14 7
The t Distributio The t distributio is a specific type of symmetric bell-shaped distributio with a lower height ad a wider spread tha the stadard ormal distributio. As the sample size becomes larger, the t distributio approaches the stadard ormal distributio. The t distributio has oly oe parameter, called the degrees of freedom (df). The mea of the t distributio is equal to 0 ad its stadard deviatio is df /( df 2) 8-15 Problems Ivolvig the t Distributio How to fid probability give t values? Table V, t.dist(t,df,1) i Excel, ad tcdf(t 1,t 2,df) i TI-84 How to fid t values give probability? Table V, or t.iv(prob-of-two-small-tails,df) i Excel. Example 8-4 Fid the value of t for 16 degrees of freedom ad.05 area i the right tail of a t distributio curve. Table V oly works o right tail 8-16 8
t Distributio Table V 8-17 Cofidece Iterval for μ Usig the t Distributio Fudametal idea Two ed poits ad a probability (cofidece) Probability comes from a kow distributio Sample mea s distributio 2 X X X ~ N(, ) ~ N(0,1) ~ T 1 / s / Fid z such that P( t T t) 0.95 (or i geeral 1 ) X Solve for from t t s / Steps to obtai cofidece iterval o μ Poit estimate ----------------- 1 Critical value ------------------- t from the distributio of T s -1 Margi of error E t Cofidece iterval ----------- ± E X X 8-18 9
Example 8-5 Accordig to a 2014 Kaiser Family Foudatio Health Beefits Survey released i 2015, the total mea cost of employer-sposored family health coverage was $16,834 per family per year, of which workers were payig a average of $4823. A radom sample of 25 workers from New York City who have employer-provided health isurace coverage paid a average premium of $6600 for family health isurace coverage with a stadard deviatio of $800. Make a 95% cofidece iterval for the curret average premium paid for family health isurace coverage by all workers i New York City who have employer-provided health isurace coverage. Assume that the distributio of premiums paid for family health isurace coverage by all workers i New York City who have employer-provided health isurace coverage is ormally distributed. Solutio Summary statistics: = 25, X = 6600, s = 800 & α = 0.05 Poit estimate: 6600 Critical value of 95%: 2.064 (from T 24 ) Margi of error: 2.064 x 800/5 = 330.24 95% C. I. o μ is 6600 ± 330.24 8-19 Example 8-6 Sixty-four radomly selected adults who buy books for geeral readig were asked how much they usually sped o books per year. The sample produced a mea of $1450 ad a stadard deviatio of $300 for such aual expeses. Determie a 99% cofidece iterval for the correspodig populatio mea. Solutio Summary statistics: = 64, X = 1450, s = 300 & α = 0.01 Poit estimate: 1450 Critical value: z = 2.58 (or as book t = 2.656) Margi or error: 2.58 x 300/8 = 96.75 (or 99.6 usig t = 2.656) 99% C. I. o μ is 1450 ± 96.75 (99.60) 8-20 10
8.4 Estimatio of Populatio Proportio p Poit estimate of populatio proportio p: x Samplig distributio of pˆ Approximate ormal distributio with Mea = p pq Stadard deviatio = Estimate of the stadard deviatio Idea of cofidece iterval pq ˆˆ pˆ p pˆ N( p, ) N(0,1) pq ˆˆ Steps for cofidece iterval o p x Poit estimate pˆ Critical value z from Z distributio Margi of error E = z C. I. iterval o p ˆp E pq ˆˆ 8-21 pq ˆˆ x pˆ Example 8-7 Policy Iteractive of Eugee, Orego, coducted a study i April 2014 for the Ceter for a New America Dream that icluded a sample of 1821 America adults. Sevety five percet of the people icluded i this study said that havig basic eeds met is very or extremely importat i their visio of the America dream (www.ewdream.org). What is the poit estimate of the populatio proportio? Fid, with a 99% cofidece level, the percetage of all America adults who will say that havig basic eeds met is very or extremely importat i their visio of the America dream. What is the margi of error of this estimate? Solutio Poit estimate 0.75 Critical value of 99%: 2.58 Margi of error: 2.58 x (0.75x0.25/1821) =0.026 99% C. I. o p: 0.75 ± 0.026 8-22 11
Example 8-8 Accordig to a Gallup-Purdue Uiversity study of college graduates coducted durig February 4 to March 7, 2014, 63% of college graduates polled said that they had at least oe college professor who made them feel excited about learig (www.gallup.com). Suppose that this study was based o a radom sample of 2000 college graduates. Costruct a 97% cofidece iterval for the correspodig populatio proportio. Solutio Summary statistics: = 2000, pˆ = 0.63 Poit estimate: 0.63 Critical value: 2.17 Margi of error: 2.17 x (0.63x0.37/2000) = 0.023 97% C. I. o p: 0.63 ± 0.023 8-23 Sample Size for Estimatio of p Give the cofidece level ad the values of, the sample size that will produce a predetermied maximum of error E of the cofidece iterval estimate of p is z 2 E pq ˆ ˆ 2 pˆ I case the value of is ot kow, we ofte make the most coservatio estimate of the sample size by usig pˆ = 0.5 Or take a prelimiary sample (of arbitrarily determied size) ad calculate pˆ from this sample. The use this value to fid. pˆ 8-24 12
Example 8-9 Lombard Electroics Compay has just istalled a ew machie that makes a part that is used i clocks. The compay wats to estimate the proportio of these parts produced by this machie that are defective. The compay maager wats this estimate to be withi.02 of the populatio proportio for a 95% cofidece level. What is the most coservative estimate of the sample size that will limit the maximum error to withi.02 of the populatio proportio? Solutio 95% => z = 1.96 Take pˆ = 0.5 Therefore, = 1.96 x 0.5 x 0.5 / 0.02 2 = 2401 8-25 Techology Istructio TI 83 plus / TI 84 Data set Summary statistics Excel Data set Fuctio Cofidece 8-26 13