Introduction. Chapter 8: Estimation of Mean & Proportion. 8.1 Estimation, Point Estimate, and Interval Estimate. Point & Interval Estimates

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1 Chapter 8: Estimatio of Mea & Proportio 8.1 Estimatio, Poit Estimate, ad Iterval Estimate 8. 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 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 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 type of light bulbs? Radom sample: 0 light bulbs What is estimator? What is estimate? 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? values 1 probability Why do we eed two estimates?

2 Example of Estimatio 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%. 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 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) 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 X X ~ N (, ) ~ N (0,1) / Fid z such that P( z Z z) 0.95 (or i geeral 1 ) Solve for from X 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 X ± E

3 Fidig z for a 100(1 α)% Cofidece Level How to fid z? 8-9 Left tail of z = 1 α/ Table, Excel (ormsiv), or calculator (ivnorm) 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 5 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) = z = 1.65 E = z = /5 = = % C.I. for is Example 8. Large Sample Sample Size for the Estimatio of Mea Accordig to Moebs Services Ic., a idividual checkig accout at major U.S. baks costs the baks betwee $350 ad $450 per year (Time, November 1, 011). 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: % => z, i.e., P(Z < z) = z = E = z = = % C.I. for is 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 z E

4 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 =.58 E = 800 Stadard deviatio σ = Therefore, = z σ / E = 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)???? The t Distributio Problems Ivolvig 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 ) How to fid probability give t values? Table V, tdist(t,df,tails) i Excel, ad tcdf(t 1,t,df) i TI-83 How to fid t values give probability? Table V, or tiv(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

5 t Distributio Table V 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 X X X ~ N(, ) ~ N(0,1) ~ T 1 / s / Fid z such that P( t T t) 0.95 (or i geeral 1 ) 1 X Solve for from t t s / Steps to obtai cofidece iterval o μ Poit estimate X Critical value t from the distributio of T s -1 Margi of error E t Cofidece iterval X ± E Example 8-5 Accordig to the Kaiser Family Foudatio, U.S. workers who had employer-provided health isurace coverage paid a average premium of $419 for family health isurace coverage durig 011 (USA TODAY, October 10, 011). A radom sample of 5 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: = 5, X = 6600, s = 800 & α = 0.05 Poit estimate: 6600 Critical value of 95%:.064 (from T 4 ) Margi of error:.064 x 800/5 = % C. I. o μ is 6600 ± 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 =.58 (or as book t =.656) Margi or error:.58 x 300/8 = (or 99.6 usig t =.656) 99% C. I. o μ is 1450 ± (99.60) 8-0 5

6 8.4 Estimatio of Populatio Proportio p Poit estimate of populatio proportio p: x Samplig distributio of pˆ Approximate ormal distributio with Mea = p Stadard deviatio = pq pq ˆˆ Estimate of the stadard deviatio pq ˆˆ pˆ p pˆ N( p, ) N(0,1) pq ˆˆ x Poit estimate pˆ Critical value z from Z distributio pq ˆˆ Margi of error E = z C. I. iterval o p Idea of cofidece iterval Steps for cofidece iterval o p ˆp E 8-1 x pˆ Example 8-7 Accordig to a New York Times/CBS News poll coducted durig Jue 4-8, 011, 55% of America adults polled said that owig a home is a very importat part of the America Dream (The New York Times, Jue 30, 011). This poll was based o a sample of 979 America adults. 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 owig a home is a very importat part of the America Dream. What is the margi of error of this estimate? Solutio Summary statistics: = 979, pˆ = 0.55 Poit estimate 0.55 Critical value of 99%:.58 Margi of error:.58 x (0.55x0.45/979) = % C. I. o p: 0.55 ± Example 8-8 Accordig to a Pew Research Ceter atiowide telephoe survey of adults coducted March 15 to April 4, 011, 86% of college graduates said that college educatio was a good ivestmet (Time, May 30, 011). Suppose that this survey icluded 1450 college graduates. Costruct a 97% cofidece iterval for the correspodig populatio proportio. Solutio Summary statistics: = 1450, pˆ = 0.86 Poit estimate: 0.86 Critical value:.17 Margi of error:.17 x (0.86x0.14/1450) = % C. I. o p: 0.86 ± 0.0 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 pq ˆ ˆ E I case the value of pˆ 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

7 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.0 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.0 of the populatio proportio? Solutio 95% => z = 1.96 Take pˆ = 0.5 Therefore, = 1.96 x 0.5 x 0.5 / 0.0 = 401 Techology Istructio TI 83 plus / TI 84 Data set Summary statistics Excel Data set Fuctio Cofidece