Estimatio ad Cofidece Itervals Chapter 9 McGraw-Hill/Irwi Copyright 2010 by The McGraw-Hill Compaies, Ic. All rights reserved.
GOALS 1. Defie a poit estimate. 2. Defie level of cofidece. 3. Costruct a cofidece iterval for the populatio mea whe the populatio stadard deviatio is kow. 4. Costruct a cofidece iterval for a populatio mea whe the populatio stadard deviatio is ukow. 5. Costruct a cofidece iterval for a populatio proportio. 6. Determie the sample size for attribute ad variable samplig. 9-2
Reasos for Samplig 1. To cotact the etire populatio is too time cosumig. 2. The cost of studyig all the items i the populatio is ofte too expesive. 3. The sample results are usually adequate. 4. Certai tests are destructive. 5. Checkig all the items is physically impossible. 9-3
Poit ad Iterval Estimates A poit estimate is a sigle value (poit) derived from a sample ad used to estimate a populatio value. A cofidece iterval estimate is a rage of values costructed from sample data so that the populatio parameter is likely to occur withi that rage at a specified probability. The specified probability is called the level of cofidece. 9-4
Factors Affectig Cofidece Iterval Estimates The factors that determie the width of a cofidece iterval are: 1.The sample size,. 2.The variability i the populatio, usually σ estimated by s. 3.The desired level of cofidece. 9-5
Iterval Estimates - Iterpretatio For a 95% cofidece iterval about 95% of the similarly costructed itervals will cotai the parameter beig estimated. Also 95% of the sample meas for a specified sample size will lie withi 1.96 stadard deviatios of the hypothesized populatio 9-6
How to Obtai z value for a Give Cofidece Level The 95 percet cofidece refers to the middle 95 percet of the observatios. Therefore, the remaiig 5 percet are equally divided betwee the two tails. Followig is a portio of Appedix B.1. 9-7
Poit Estimates ad Cofidece Itervals for a Mea σ Kow x z σ sample mea z - value for a particular cofidecelevel the populatio stadard deviatio the umber of observatios i the sample 1. The width of the iterval is determied by the level of cofidece ad the size of the stadard error of the mea. 2. The stadard error is affected by two values: - Stadard deviatio - Number of observatios i the sample 9-8
Example: Cofidece Iterval for a Mea σ Kow The America Maagemet Associatio wishes to have iformatio o the mea icome of middle maagers i the retail idustry. A radom sample of 256 maagers reveals a sample mea of $45,420. The stadard deviatio of this populatio is $2,050. The associatio would like aswers to the followig questios: 1. What is the populatio mea? 2. What is a reasoable rage of values for the populatio mea? 3. What do these results mea? 9-9
Example: Cofidece Iterval for a Mea σ Kow The America Maagemet Associatio wishes to have iformatio o the mea icome of middle maagers i the retail idustry. A radom sample of 256 maagers reveals a sample mea of $45,420. The stadard deviatio of this populatio is $2,050. The associatio would like aswers to the followig questios: What is the populatio mea? I this case, we do ot kow. We do kow the sample mea is $45,420. Hece, our best estimate of the ukow populatio value is the correspodig sample statistic. The sample mea of $45,420 is a poit estimate of the ukow populatio mea. 9-10
Example: Cofidece Iterval for a Mea σ Kow The America Maagemet Associatio wishes to have iformatio o the mea icome of middle maagers i the retail idustry. A radom sample of 256 maagers reveals a sample mea of $45,420. The stadard deviatio of this populatio is $2,050. The associatio would like aswers to the followig questios: What is a reasoable rage of values for the populatio mea? Suppose the associatio decides to use the 95 percet level of cofidece: The cofidece limit are $45,169 ad $45,671 The $251 is referred to as the margi of error 9-11
Example: Cofidece Iterval for a Mea σ Kow The America Maagemet Associatio wishes to have iformatio o the mea icome of middle maagers i the retail idustry. A radom sample of 256 maagers reveals a sample mea of $45,420. The stadard deviatio of this populatio is $2,050. The associatio would like aswers to the followig questios: What do these results mea, i.e. what is the iterpretatio of the cofidece limits $45,169 ad $45,671? If we select may samples of 256 maagers, ad for each sample we compute the mea ad the costruct a 95 percet cofidece iterval, we could expect about 95 percet of these cofidece itervals to cotai the populatio mea. Coversely, about 5 percet of the itervals would ot cotai the populatio mea aual icome, µ 9-12
Populatio Stadard Deviatio (σ) Ukow I most samplig situatios the populatio stadard deviatio (σ) is ot kow. Below are some examples where it is ulikely the populatio stadard deviatios would be kow. 1. The Dea of the Busiess College wats to estimate the mea umber of hours full-time studets work at payig jobs each week. He selects a sample of 30 studets, cotacts each studet ad asks them how may hours they worked last week. 2. The Dea of Studets wats to estimate the distace the typical commuter studet travels to class. She selects a sample of 40 commuter studets, cotacts each, ad determies the oe-way distace from each studet s home to the ceter of campus. 9-13 3. The Director of Studet Loas wats to kow the mea amout owed o studet loas at the time of his/her graduatio. The director selects a sample of 20 graduatig studets ad cotacts each to fid the iformatio.
Characteristics of the t-distributio 1. It is, like the z distributio, a cotiuous distributio. 2. It is, like the z distributio, bell-shaped ad symmetrical. 3. There is ot oe t distributio, but rather a family of t distributios. All t distributios have a mea of 0, but their stadard deviatios differ accordig to the sample size,. 4. The t distributio is more spread out ad flatter at the ceter tha the stadard ormal distributio As the sample size icreases, however, the t distributio approaches the stadard ormal distributio 9-14
9-15 Comparig the z ad t Distributios whe is small, 95% Cofidece Level
Cofidece Iterval Estimates for the Mea Use Z-distributio If the populatio stadard deviatio is kow or the sample is greater tha 30. Use t-distributio If the populatio stadard deviatio is ukow ad the sample is less tha 30. 9-16
9-17 Whe to Use the z or t Distributio for Cofidece Iterval Computatio
Cofidece Iterval for the Mea Example usig the t-distributio A tire maufacturer wishes to ivestigate the tread life of its tires. A sample of 10 tires drive 50,000 miles revealed a sample mea of 0.32 ich of tread remaiig with a stadard deviatio of 0.09 ich. Costruct a 95 percet cofidece iterval for the populatio mea. Would it be reasoable for the maufacturer to coclude that after 50,000 miles the populatio mea amout of tread remaiig is 0.30 iches? Give i the problem: x s X 10 0.32 0.09 Compute t - dist.(sice t / 2, 1 the C.I. s usig the is ukow) 9-18
9-19 Studet s t-distributio Table
Cofidece Iterval Estimates for the Mea Usig Miitab The maager of the Ilet Square Mall, ear Ft. Myers, Florida, wats to estimate the mea amout spet per shoppig visit by customers. A sample of 20 customers reveals the followig amouts spet. 9-20
Cofidece Iterval Estimates for the Mea By Formula 9-21 Compute the C.I. usig the t - dist.(sice X X The edpoits of the cofideceiterval are $45.13ad $53.57 Coclude : It is reasoable that the The t 49.35 49.35 49.35 / 2, 1 t.05/ 2,20 value 9.01 t.025,19 20 9.01 2.093 20 4.22 of s coclude that the 1 s is ukow) populatio mea could be $50. $60is ot i the cofideceiterval. Hece, we populatio mea is ulikely to be $60.
9-22 Cofidece Iterval Estimates for the Mea Usig Miitab
9-23 Cofidece Iterval Estimates for the Mea Usig Excel
A Cofidece Iterval for a Proportio (π) The examples below illustrate the omial scale of measuremet. 1. The career services director at Souther Techical Istitute reports that 80 percet of its graduates eter the job market i a positio related to their field of study. 2. A compay represetative claims that 45 percet of Burger Kig sales are made at the drive-through widow. 3. A survey of homes i the Chicago area idicated that 85 percet of the ew costructio had cetral air coditioig. 4. A recet survey of married me betwee the ages of 35 ad 50 foud that 63 percet felt that both parters should ear a livig. 9-24
Usig the Normal Distributio to Approximate the Biomial Distributio To develop a cofidece iterval for a proportio, we eed to meet the followig assumptios. 1. The biomial coditios, discussed i Chapter 6, have bee met. Briefly, these coditios are: a. The sample data is the result of couts. b. There are oly two possible outcomes. c. The probability of a success remais the same from oe trial to the ext. d. The trials are idepedet. This meas the outcome o oe trial does ot affect the outcome o aother. 2. The values π ad (1-π) should both be greater tha or equal to 5. This coditio allows us to ivoke the cetral limit theorem ad employ the stadard ormal distributio, that is, z, to complete a cofidece iterval. 9-25
9-26 Cofidece Iterval for a Populatio Proportio - Formula
Cofidece Iterval for a Populatio Proportio- Example 9-27 The uio represetig the Bottle Blowers of America (BBA) is cosiderig a proposal to merge with the Teamsters Uio. Accordig to BBA uio bylaws, at least three-fourths of the uio membership must approve ay merger. A radom sample of 2,000 curret BBA members reveals 1,600 pla to vote for the merger proposal. What is the estimate of the populatio proportio? Develop a 95 percet cofidece iterval for the populatio proportio. Basig your decisio o this sample iformatio, ca you coclude that the ecessary proportio of BBA members favor the merger? Why? First, compute the sample proportio: p C.I. x Compute the 95%C.I. p 1,600 2000 z 0.80 / 2 (0.782, 1.96 0.80 p(1 0.818) p).80(1.80) 2,000.80.018 Coclude : The merger proposal will likely pass because the iterval estimateicludes values greater tha 75percet of the uio membership.
Fiite-Populatio Correctio Factor A populatio that has a fixed upper boud is said to be fiite. For a fiite populatio, where the total umber of objects is N ad the size of the sample is, the followig adjustmet is made to the stadard errors of the sample meas ad the proportio: However, if /N <.05, the fiite-populatio correctio factor may be igored. Stadard Error of the Mea Stadard Error of the Proportio x N N 1 p p(1 p) N N 1 9-28
Effects o FPC whe /N Chages Observe that FPC approaches 1 whe /N becomes smaller 9-29
Cofidece Iterval Formulas for Estimatig Meas ad Proportios with Fiite Populatio Correctio C.I. for the Mea ( ) C.I. for the Mea ( ) X z N N 1 X t s N N 1 C.I. for the Proportio ( ) p z p(1 p) N N 1 9-30
CI for Mea with FPC - Example 9-31 There are 250 families i Scadia, Pesylvaia. A radom sample of 40 of these families revealed the mea aual church cotributio was $450 ad the stadard deviatio of this was $75. Could the populatio mea be $445 or $425? 1. What is the populatio mea? What is the best estimate of the populatio mea? 2. Discuss why the fiitepopulatio correctio factor should be used. Give i Problem: N 250 40 s - $75 Sice /N = 40/250 = 0.16, the fiite populatio correctio factor must be used. The populatio stadard deviatio is ot kow therefore use the t- distributio (may use the z-dist sice >30) Use the formula below to compute the cofidece iterval: X t s N N 1
CI For Mea with FPC - Example X t s N N 1 $450 t.10/ 2,40 1 $75 40 250 250 40 1 $450 $75 1.685 40 250 250 40 1 $450 $19.98.8434 $450 $18.35 ($431.65, $468.35) It is likely that the populatio mea is more tha $431.65but less tha $468.35. 9-32 To put it aother way, could the populatio mea be $445?Yes, but it is ot likely that it is $425because the value $445is withi the cofidece iterval ad $425is ot withi the cofideceiterval.
Selectig a Appropriate Sample Size There are 3 factors that determie the size of a sample, oe of which has ay direct relatioship to the size of the populatio. The level of cofidece desired. The margi of error the researcher will tolerate. The variatio i the populatio beig Studied. 9-33
Sample Size for Estimatig the Populatio Mea z E 2 9-34
Sample Size Determiatio for a Variable-Example A studet i public admiistratio wats to determie the mea amout members of city coucils i large cities ear per moth as remueratio for beig a coucil member. The error i estimatig the mea is to be less tha $100 with a 95 percet level of cofidece. The studet foud a report by the Departmet of Labor that estimated the stadard deviatio to be $1,000. What is the required sample size? Give i the problem: E, the maximum allowable error, is $100 The value of z for a 95 percet level of cofidece is 1.96, The estimate of the stadard deviatio is $1,000. z E (1.96)($1,000) $100 (19.6) 2 384.16 2 2 385 9-35
Sample Size Determiatio for a Variable- Aother Example A cosumer group would like to estimate the mea mothly electricity charge for a sigle family house i July withi $5 usig a 99 percet level of cofidece. Based o similar studies the stadard deviatio is estimated to be $20.00. How large a sample is required? (2.58)(20) 5 2 107 9-36
Sample Size for Estimatig a Populatio Proportio p(1 p) Z E 2 where: is the size of the sample z is the stadard ormal value correspodig to the desired level of cofidece E is the maximum allowable error 9-37
Aother Example The America Keel Club wated to estimate the proportio of childre that have a dog as a pet. If the club wated the estimate to be withi 3% of the populatio proportio, how may childre would they eed to cotact? Assume a 95% level of cofidece ad that the club estimated that 30% of the childre have a dog as a pet. (.30)(.70) 1.96.03 2 897 9-38
Aother Example A study eeds to estimate the proportio of cities that have private refuse collectors. The ivestigator wats the margi of error to be withi.10 of the populatio proportio, the desired level of cofidece is 90 percet, ad o estimate is available for the populatio proportio. What is the required sample size? (.5)(1 69cities.5) 1.65.10 2 68.0625 9-39