Chapter 8 Student Lecture Notes 8-1

Transcription

1 Chapter 8 tudet Lecture Notes 8-1 Basic Busiess tatistics (9 th Editio) Chapter 8 Cofidece Iterval Estimatio 004 Pretice-Hall, Ic. Chap 8-1 Chapter Topics Estimatio Process Poit Estimates Iterval Estimates Cofidece Iterval Estimatio for the Mea ( σ Kow) Determiig ample ize Cofidece Iterval Estimatio for the Mea ( σ Ukow) 004 Pretice-Hall, Ic. Chap 8- Chapter Topics (cotiued) Cofidece Iterval Estimatio ad ample ize Determiatio for the Proportio Cofidece Iterval Estimatio for Populatio Total Cofidece Iterval Estimatio for Total Differece i the Populatio Estimatio ad ample ize Determiatio for Fiite Populatio (CD-ROM Topic) Cofidece Iterval Estimatio ad Ethical Issues 004 Pretice-Hall, Ic. Chap 8-3 tatistics for Maagers Usig Microsoft Excel, /e 1999 Pretice-Hall, Ic.

2 Chapter 8 tudet Lecture Notes 8- Estimatio Process Populatio Mea,, is ukow Radom ample Mea X = 50 I am 95% cofidet that is betwee 40 & 60. ample 004 Pretice-Hall, Ic. Chap 8-4 Poit Estimates Estimate Populatio Parameters Mea Proportio Variace Differece p σ 1 with ample tatistics X X P X Pretice-Hall, Ic. Chap 8-5 Iterval Estimates Provide Rage of Values Take ito cosideratio variatio i sample statistics from sample to sample Based o observatio from 1 sample Give iformatio about closeess to ukow populatio parameters tated i terms of level of cofidece Never 100% sure 004 Pretice-Hall, Ic. Chap 8-6 tatistics for Maagers Usig Microsoft Excel, /e 1999 Pretice-Hall, Ic.

3 Chapter 8 tudet Lecture Notes 8-3 Cofidece Iterval Estimates Cofidece Itervals Mea Proportio σ Kow σ Ukow 004 Pretice-Hall, Ic. Chap 8-7 Cofidece Iterval for ( σ Kow) Assumptios Critical Value tadard Error Populatio stadard deviatio is kow Populatio is ormally distributed If populatio is ot ormal, use large sample Cofidece Iterval Estimate σ X Zα/ X + Zα/ e= Z σ α / is called the samplig error or margi of error 004 Pretice-Hall, Ic. Chap 8-8 σ Elemets of Cofidece Iterval Estimatio Level of Cofidece Cofidece that the iterval will cotai the ukow populatio parameter Precisio (Rage) Closeess to the ukow parameter Cost Cost required to obtai a sample of size 004 Pretice-Hall, Ic. Chap 8-9 tatistics for Maagers Usig Microsoft Excel, /e 1999 Pretice-Hall, Ic.

4 Chapter 8 tudet Lecture Notes 8-4 Level of Cofidece ( ) Deoted by α % A Relative Frequecy Iterpretatio I the log ru, 100( 1 α )% of all the cofidece itervals that ca be costructed will cotai (bracket) the ukow parameter A pecific Iterval Will Either Cotai or Not Cotai the Parameter 004 Pretice-Hall, Ic. Chap 8-10 Iterval ad Level of Cofidece amplig Distributio of the _ Mea Z α / σ X Itervals exted from X Zσ X to + X Zσ X α / 1 α = X α / Cofidece Itervals + Z α / σ X X ( ) 1 α 100% of itervals costructed cotai ; 100 α% do ot. 004 Pretice-Hall, Ic. Chap 8-11 Example A radom sample of 15 stocks traded o the NADAQ market showed a average of 1500 shares traded. From past experiece, it is believed that the populatio stadard deviatio of shares traded is ad the shares traded are very closed to a ormal distributio. Costruct a 99% cofidece iterval for the average shares traded o the NADAQ market. Iterpret your result. PHtat output Cofidece Iterval Estimate for the Mea Populatio tadard Deviatio ample Mea ample ize 15 Cofidece Level 99% tadard Error of the Mea Z Value Iterval Half Width Iterval Lower Limit Iterval Upper Limit The 99% CI for the populatio mea: < < Pretice-Hall, Ic. Chap 8-1 tatistics for Maagers Usig Microsoft Excel, /e 1999 Pretice-Hall, Ic.

5 Chapter 8 tudet Lecture Notes 8-5 Example: Iterpretatio (cotiued) If all possible samples of size 15 are take ad the correspodig 99% cofidece itervals are costructed, 99% of the cofidece itervals that are costructed will cotai the true ukow populatio mea. We are 99% cofidet that the populatio average umber of shares traded o the NADAQ is betwee ad Usig the cofidece iterval method o repeated samplig, the probability that we will have costructed a cofidece iterval that will cotai the ukow populatio mea is 99%. 004 Pretice-Hall, Ic. Chap 8-13 Obtaiig Cofidece Iterval i PHtat PHtat Cofidece Iterval Estimates for the Mea, igma Kow 004 Pretice-Hall, Ic. Chap 8-14 Factors Affectig Iterval Width (Precisio) Data Variatio Measured by σ ample ize σ σ = X Level of Cofidece 100( 1 α )% Itervals Exted from X - Zσ to X + Z σ x x T/Maker Co. 004 Pretice-Hall, Ic. Chap 8-15 tatistics for Maagers Usig Microsoft Excel, /e 1999 Pretice-Hall, Ic.

6 Chapter 8 tudet Lecture Notes 8-6 Determiig ample ize (Cost) Too Big: Requires more resources Too small: Wo t do the job 004 Pretice-Hall, Ic. Chap 8-16 Determiig ample ize for Mea What sample size is eeded to be 90% cofidet of beig correct withi ± 5? A pilot study suggested that the stadard deviatio is 45. ( ) Z σ = = = Error 5 Roud Up 004 Pretice-Hall, Ic. Chap 8-17 Determiig ample ize for Mea i PHtat PHtat ample ize Determiatio for the Mea Example i Excel preadsheet ample ize Determiatio Microsoft Excel Worksheet Data Populatio tadard Deviatio 45 amplig Error 5 Cofidece Level 90% Itemediate Calculatios Z Value Calculated ample ize Result ample ize Needed Pretice-Hall, Ic. Chap 8-18 tatistics for Maagers Usig Microsoft Excel, /e 1999 Pretice-Hall, Ic.

7 Chapter 8 tudet Lecture Notes 8-7 Cofidece Iterval for ( σ Ukow) tadard Error Assumptios Margi of Error Populatio stadard deviatio is ukow Populatio is ormally distributed If populatio is ot ormal, use large sample Use tudet s t Distributio Cofidece Iterval Estimate X tα X + t /, 1 α/, Pretice-Hall, Ic. Chap 8-19 tudet s t Distributio tadardized Normal Bell-haped ymmetric Fatter Tails t (df = 13) t (df = 5) 0 Z t 004 Pretice-Hall, Ic. Chap 8-0 tudet s t Table Upper Tail Area df Let: = 3 df = -1 = α =.10 α/ =.05 α/ = t Values 0.90 t 004 Pretice-Hall, Ic. Chap 8-1 tatistics for Maagers Usig Microsoft Excel, /e 1999 Pretice-Hall, Ic.

8 Chapter 8 tudet Lecture Notes 8-8 Example A radom sample of = 5 has X = 50 ad = 8. et up a 95% cofidece iterval estimate for. X tα/, 1 X+ tα/, We are 95% cofidet that the ukow true populatio mea is somewhere betwee ad Pretice-Hall, Ic. Chap 8- Cofidece Iterval for ( σ Ukow) i PHtat PHtat Cofidece Iterval Estimate for the Mea, igma Ukow Example i Excel preadsheet Cofidece Iterval Estimate for the Mea Microsoft Excel Worksheet Data ample tadard Deviatio 8 ample Mea 50 ample ize 5 Cofidece Level 95% Itermediate Calculatios tadard Error of the Mea 1.6 Degrees of Freedom 4 t Value Iterval Half Width Cofidece Iterval Iterval Lower Limit Iterval Upper Limit Pretice-Hall, Ic. Chap 8-3 Cofidece Iterval Estimate for Proportio tadard Error Assumptios Margi of Error Two categorical outcomes Populatio follows biomial distributio Normal approximatio ca be used if p 5 ad ( 1 p) 5 Cofidece Iterval Estimate p ( 1 p) p ( 1 p) p Zα/ p p + Zα/ 004 Pretice-Hall, Ic. Chap 8-4 tatistics for Maagers Usig Microsoft Excel, /e 1999 Pretice-Hall, Ic.

9 Chapter 8 tudet Lecture Notes 8-9 Example A radom sample of 400 voters showed that 3 preferred Cadidate A. et up a 95% cofidece iterval estimate for p. ps( 1 ps) ps( 1 ps) ps Zα/ p ps + Zα/.08( 1.08 ).08( 1.08) p p.107 We are 95% cofidet that the proportio of voters who prefer Cadidate A is somewhere betwee ad Pretice-Hall, Ic. Chap 8-5 Cofidece Iterval Estimate for Proportio i PHtat PHtat Cofidece Iterval Estimate for the Proportio Example i Excel preadsheet Cofidece Iterval Estimate for the Mea Microsoft Excel Worksheet Data ample ize 400 Number of uccesses 3 Cofidece Level 95% Itermediate Calculatios ample Proportio 0.08 Z Value tadard Error of the Proportio Iterval Half Width Cofidece Iterval Iterval Lower Limit Iterval Upper Limit Pretice-Hall, Ic. Chap 8-6 Determiig ample ize for Proportio What sample size is eeded to be withi ±5% with 90% cofidece if past studies show about 30% are defective? ( 1 p) ( 0.3)( 0.7) Z p = = Error 0.05 = Roud Up 004 Pretice-Hall, Ic. Chap 8-7 tatistics for Maagers Usig Microsoft Excel, /e 1999 Pretice-Hall, Ic.

10 Chapter 8 tudet Lecture Notes 8-10 Determiig ample ize for Proportio i PHtat PHtat ample ize Determiatio for the Proportio Example i Excel preadsheet ample ize Determiatio Microsoft Excel Worksheet Data Estimate of True Proportio 0.3 amplig Error 0.05 Cofidece Level 90% Itermediate Calculatios Z Value Calculated ample ize Result ample ize Needed Pretice-Hall, Ic. Chap 8-8 Cofidece Iterval for Populatio Total Amout Poit Estimate NX Cofidece Iterval Estimate NX ± N t ( α /, 1) ( N ) ( N 1) 004 Pretice-Hall, Ic. Chap 8-9 Cofidece Iterval for Populatio Total: Example A auditor is faced with a populatio of 1000 vouchers ad wishes to estimate the total value of the populatio of vouchers. A sample of 50 vouchers is selected with the average voucher amout of $ , stadard deviatio of $73.6. et up the 95% cofidece iterval estimate of the total amout for the populatio of vouchers. 004 Pretice-Hall, Ic. Chap 8-30 tatistics for Maagers Usig Microsoft Excel, /e 1999 Pretice-Hall, Ic.

11 Chapter 8 tudet Lecture Notes 8-11 ( α /, 1) Example olutio N = 1000 = 50 X = $ = $73.6 NX ± N t ( N ) ( N 1) ( 1000)( ) ( 1000)(.0096) = 1,076,390 ± 75, = ± The 95% cofidece iterval for the populatio total amout of the vouchers is betwee 1,000, ad 1,15, Pretice-Hall, Ic. Chap 8-31 Example olutio i PHtat PHtat Cofidece Itervals Estimate for the Populatio Total Excel preadsheet for the Voucher Example Microsoft Excel Worksheet 004 Pretice-Hall, Ic. Chap 8-3 Cofidece Iterval for Total Differece i the Populatio Poit Estimate Di i= 1 ND where D = is the sample average differece Cofidece Iterval Estimate N D ND ± N ( tα /, 1) N 1 where D = ( D ) i D i= 1 1 ( ) ( ) 004 Pretice-Hall, Ic. Chap 8-33 tatistics for Maagers Usig Microsoft Excel, /e 1999 Pretice-Hall, Ic.

12 Chapter 8 tudet Lecture Notes 8-1 Estimatio for Fiite Populatio (CD-ROM Topic) amples are elected Without Replacemet Cofidece iterval for the mea ( σ ukow) X Cofidece iterval for proportio p ± t α /, 1 ± Z α / ( N ) ( N 1) ( 1 ) ( ) ( N 1) p p N 004 Pretice-Hall, Ic. Chap 8-34 ample ize ( ) Determiatio for Fiite Populatio (CD-ROM Topic) amples are elected Without Replacemet = Whe estimatig the mea Zα /σ 0 = Whe estimatig the proportio N ( N ) e α / 0 = ( 1 ) Z p p e 004 Pretice-Hall, Ic. Chap 8-35 Ethical Issues Cofidece Iterval (Reflects amplig Error) hould Always Be Reported Alog with the Poit Estimate The Level of Cofidece hould Always Be Reported The ample ize hould Be Reported A Iterpretatio of the Cofidece Iterval Estimate hould Also Be Provided 004 Pretice-Hall, Ic. Chap 8-36 tatistics for Maagers Usig Microsoft Excel, /e 1999 Pretice-Hall, Ic.

13 Chapter 8 tudet Lecture Notes 8-13 Chapter ummary Illustrated Estimatio Process Discussed Poit Estimates Addressed Iterval Estimates Discussed Cofidece Iterval Estimatio for the Mea ( σ Kow) Addressed Determiig ample ize Discussed Cofidece Iterval Estimatio for the Mea ( Ukow) σ 004 Pretice-Hall, Ic. Chap 8-37 Chapter ummary (cotiued) Discussed Cofidece Iterval Estimatio for the Proportio Addressed Cofidece Iterval Estimatio for Populatio Total Discussed Cofidece Iterval Estimatio for Total Differece i the Populatio Addressed Estimatio ad ample ize Determiatio for Fiite Populatio (CD-ROM Topic) Addressed Cofidece Iterval Estimatio ad Ethical Issues 004 Pretice-Hall, Ic. Chap 8-38 tatistics for Maagers Usig Microsoft Excel, /e 1999 Pretice-Hall, Ic.