Chapter 7 Student Lecture Notes 7-1

Transcription

1 Chapter 7 Studet Lecture otes 7-1 Basic Busiess Statistics (9 th Editio) Chapter 7 Samplig Distributios 24 Pretice-Hall, Ic. Chap 7-1 Chapter Topics Samplig Distributio of the Mea The Cetral Limit Theorem Samplig Distributio of the Proportio Samplig from Fiite Populatio 24 Pretice-Hall, Ic. Chap 7-2 Why Study Samplig Distributios Sample Statistics are Used to Estimate Populatio Parameters E.g., 5 estimates the populatio mea µ Problem: Differet Samples Provide Differet Estimates Large sample gives better estimate; large sample costs more How good is the estimate? Approach to Solutio: Theoretical Basis is Samplig Distributio 24 Pretice-Hall, Ic. Chap 7-3

2 Chapter 7 Studet Lecture otes 7-2 Samplig Distributio Theoretical Probability Distributio of a Sample Statistic Sample Statistic is a Radom Variable Sample mea, sample proportio Results from Takig All Possible Samples of the Same Size 24 Pretice-Hall, Ic. Chap 7-4 Developig Samplig Distributios Suppose There is a Populatio Populatio Size 4 B Radom Variable,, is Age of Idividuals Measured i Years Values of : 18, 2, 22, 24 A C D 24 Pretice-Hall, Ic. Chap 7-5 i i 1 µ ( i µ ) i 1 Developig Samplig Distributios 236 P() A B C D (18) (2) (22) (24) Uiform Distributio (cotiued) Summary Measures for the Populatio Distributio 24 Pretice-Hall, Ic. Chap 7-6

3 Chapter 7 Studet Lecture otes 7-3 All Possible Samples of Size 2 1 st 2 d Observatio Obs ,18 18,2 18,22 18,24 2 2,18 2,2 2,22 2, ,18 22,2 22,22 22, ,18 24,2 24,22 24, Samples Take with Replacemet Developig Samplig Distributios (cotiued) 16 Sample Meas 1st 2d Observatio Obs Pretice-Hall, Ic. Chap 7-7 Samplig Distributio of All Sample Meas 16 Sample Meas 1st 2d Observatio Obs Developig Samplig Distributios Pretice-Hall, Ic. Chap 7-8 ( ) P Sample Meas Distributio (cotiued) _ i i L+ 24 µ ( i µ ) i 1 Developig Samplig Distributios 2 ( 18 21) + ( 19 21) + L+ ( 24 21) (cotiued) Summary Measures of Samplig Distributio Pretice-Hall, Ic. Chap 7-9

4 Chapter 7 Studet Lecture otes 7-4 Comparig the Populatio with Its Samplig Distributio Populatio Sample Meas Distributio 4 2 µ µ P( ) P( ) A B C D (18) (2) (22) (24) _ 24 Pretice-Hall, Ic. Chap 7-1 Properties of Summary Measures µ µ I.e., is ubiased Stadard Error (Stadard Deviatio) of the Samplig Distributio is Less Tha the Stadard Error of Other Ubiased Estimators For Samplig with Replacemet or without Replacemet from Large or Ifiite Populatios: As icreases, decreases 24 Pretice-Hall, Ic. Chap 7-11 f ( ) Ubiasedess ( µ µ ) Ubiased µ µ 24 Pretice-Hall, Ic. Chap 7-12

5 Chapter 7 Studet Lecture otes 7-5 f ( ) Samplig Distributio of Media Less Variability Stadard Error (Stadard Deviatio) of the Samplig Distributio is Less Tha the Stadard Error of Other Ubiased Estimators Samplig Distributio of Mea µ 24 Pretice-Hall, Ic. Chap 7-13 f ( ) Effect of Large Sample For samplig with replacemet: As icreases, decreases Smaller sample size Larger sample size µ 24 Pretice-Hall, Ic. Chap 7-14 Whe the Populatio is ormal Cetral Tedecy µ µ Variatio Populatio Distributio 1 µ 5 Samplig Distributios µ 5 24 Pretice-Hall, Ic. Chap 7-15

6 Chapter 7 Studet Lecture otes 7-6 Cetral Tedecy µ µ Variatio Whe the Populatio is ot ormal Populatio Distributio 1 µ 5 Samplig Distributios µ 5 24 Pretice-Hall, Ic. Chap 7-16 Cetral Limit Theorem As Sample Size Gets Large Eough Samplig Distributio Becomes Almost ormal Regardless of Shape of Populatio 24 Pretice-Hall, Ic. Chap 7-17 How Large is Large Eough? For Most Distributios, >3 For Fairly Symmetric Distributios, >15 For ormal Distributio, the Samplig Distributio of the Mea is Always ormally Distributed Regardless of the Sample Size This is a property of samplig from a ormal populatio distributio ad is OT a result of the cetral limit theorem 24 Pretice-Hall, Ic. Chap 7-18

7 Chapter 7 Studet Lecture otes 7-7 Example: Samplig Distributio µ P 7.8 < < 8? ( ) P( 7.8 < < 8) P < < 2/ 25 2/ 25 P.5 < <.5 83 µ ( ) Stadardized ormal Distributio µ µ.5 24 Pretice-Hall, Ic. Chap 7-19 Populatio Proportio Categorical Variable ( p) E.g., Geder, Voted for Bush, College Degree Proportio of Populatio Havig a Characteristic ( p) Sample Proportio Provides a Estimate umber of successes ps sample size If Two Outcomes, Has a Biomial Distributio Possess or do ot possess characteristic 24 Pretice-Hall, Ic. Chap 7-2 Samplig Distributio of Sample Proportio Approximated by ormal Distributio p 5 ( p) 1 5 Mea: µ p S p Stadard error: ( 1 p) Samplig Distributio f(p s ) p populatio proportio p p S 24 Pretice-Hall, Ic. Chap 7-21 p s

8 Chapter 7 Studet Lecture otes 7-8 Stadardizig Samplig Distributio of Proportio ps µ p p S S p p ps ( 1 p) Samplig Distributio Stadardized ormal Distributio 1 p S µ p S ps µ 24 Pretice-Hall, Ic. Chap 7-22 Example: ( ) 2 p.4 P p S <.43? ps µ p.43.4 S P( ps <.43) P < P( <.87).878 p.4( 1.4 S ) 2 Stadardized Samplig Distributio ormal Distributio 1 p S µ ps p 24 Pretice-Hall, Ic. S Chap 7-23 Samplig from Fiite Populatio (CD ROM Topic) Modify Stadard Error if Sample Size () is Large Relative to Populatio Size ( ) >.5 or / >.5 Use Fiite Populatio Correctio Factor (FPC) Stadard Error with FPC 1 p( 1 p) P S 1 24 Pretice-Hall, Ic. Chap 7-24

9 Chapter 7 Studet Lecture otes 7-9 Chapter Summary Discussed Samplig Distributio of the Sample Mea Described the Cetral Limit Theorem Discussed Samplig Distributio of the Sample Proportio Described Samplig from Fiite Populatios 24 Pretice-Hall, Ic. Chap 7-25