Statistics for Managers Using Microsoft Excel Chapter 7 Confidence Interval Estimation

Transcription

1 Statistics for Maagers Usig Microsoft Excel Chapter 7 Cofidece Iterval Estimatio 1999 Pretice-Hall, Ic. Chap. 7-1

2 Chapter Topics Cofidece Iterval Estimatio for the Mea (s Kow) Cofidece Iterval Estimatio for the Mea (s Ukow) Cofidece Iterval Estimatio for the Proportio The Situatio of Fiite Populatios Sample Size Estimatio 1999 Pretice-Hall, Ic. Chap. 7-2

3 Estimatio Process Populatio Mea, m, is ukow Radom Sample Mea = 50 I am 95% cofidet that m is betwee 40 & 60. Sample 1999 Pretice-Hall, Ic. Chap. 7-3

4 Populatio Parameters Estimated Estimate Populatio Parameter... Mea m with Sample Statistic _ Proportio p p s Variace s 2 Differece s 2 m - m 1 2 x - x Pretice-Hall, Ic. Chap. 7-4

5 Cofidece Iterval Estimatio Provides Rage of Values Based o Observatios from 1 Sample Gives Iformatio about Closeess to Ukow Populatio Parameter Stated i terms of Probability Never 100% Sure 1999 Pretice-Hall, Ic. Chap. 7-5

6 Elemets of Cofidece Iterval Estimatio A Probability That the Populatio Parameter Falls Somewhere Withi the Iterval. Cofidece Iterval Sample Statistic Cofidece Limit (Lower) Cofidece Limit (Upper) 1999 Pretice-Hall, Ic. Chap. 7-6

7 Cofidece Limits for Populatio Mea Parameter = Statistic ± Its Error m Error m = Error = m Z s m Error s Error Z s x m Zs T/Maker Co Pretice-Hall, Ic. Chap. 7-7

8 Cofidece Itervals Zs Z s s _ x m s 645s x m 1. 90% Samples x _ m 1.96s m x s x 95% Samples m 2. 58s x m 2. 58s 99% Samples 1999 Pretice-Hall, Ic. Chap. 7-8 x

9 Level of Cofidece Probability that the ukow populatio parameter falls withi the iterval Deoted (1 - a) % = level of cofidece e.g. 90%, 95%, 99% a Is Probability That the Parameter Is Not Withi the Iterval 1999 Pretice-Hall, Ic. Chap. 7-9

10 Itervals & Level of Cofidece Samplig Distributio of the Mea Itervals Exted from Zs to Zs a /2 m s _ x 1 - a a /2 m Cofidece Itervals _ (1 - a) % of Itervals Cotai m. a % Do Not Pretice-Hall, Ic. Chap. 7-10

11 Factors Affectig Iterval Width Data Variatio measured by s Sample Size s / s Level of Cofidece (1 - a) Itervals Exted from - Zs x to + Z s x T/Maker Co Pretice-Hall, Ic. Chap. 7-11

12 Cofidece Iterval Estimates Cofidece Itervals Mea Proportio s Kow s Ukow Fiite Populatio 1999 Pretice-Hall, Ic. Chap. 7-12

13 Cofidece Itervals (s Kow) Assumptios Populatio Stadard Deviatio Is Kow Populatio Is Normally Distributed If Not Normal, use large samples Cofidece Iterval Estimate s Z a / 2 m Z / a 2 s 1999 Pretice-Hall, Ic. Chap. 7-13

14 Cofidece Iterval Estimates Cofidece Itervals Mea Proportio s Kow s Ukow Fiite Populatio 1999 Pretice-Hall, Ic. Chap. 7-14

15 Cofidece Itervals (s Ukow) Assumptios Populatio Stadard Deviatio Is Ukow Populatio Must Be Normally Distributed Use Studet s t Distributio Cofidece Iterval Estimate t a / 2, 1 S m t a / 2, 1 S 1999 Pretice-Hall, Ic. Chap. 7-15

16 Studet s t Distributio Stadard Normal Bell-Shaped Symmetric Fatter Tails t (df = 13) t (df = 5) 0 Z t 1999 Pretice-Hall, Ic. Chap. 7-16

17 Degrees of Freedom (df) Number of Observatios that Are Free to Vary After Sample Mea Has Bee Calculated Example Mea of 3 Numbers Is 2 1 = 1 (or Ay Number) 2 = 2 (or Ay Number) 3 = 3 (Caot Vary) Mea = 2 degrees of freedom = -1 = 3-1 = Pretice-Hall, Ic. Chap. 7-17

18 Studet s t Table Upper Tail Area df a / 2 Assume: = 3 = - 1 = 2 a =.10 a/2 =.05 df t t Values Pretice-Hall, Ic. Chap

19 Example: Iterval Estimatio s Ukow A radom sample of = 25 has = 50 ad s = 8. Set up a 95% cofidece iterval estimate for m. S S ta / 2, 1 m ta / 2, m 25. m Pretice-Hall, Ic. Chap. 7-19

20 Cofidece Iterval Estimates Cofidece Itervals Mea Proportio s Kow s Ukow Fiite Populatio 1999 Pretice-Hall, Ic. Chap. 7-20

21 Estimatio for Fiite Populatios Assumptios Sample Is Large Relative to Populatio / N >.05 Use Fiite Populatio Correctio Factor Cofidece Iterval (Mea, s Ukow) t a / 2, 1 S N N 1 m t a/ 2, 1 S N N Pretice-Hall, Ic. Chap. 7-21

22 Cofidece Iterval Estimates Cofidece Itervals Mea Proportio s Kow s Ukow Fiite Populatio 1999 Pretice-Hall, Ic. Chap. 7-22

23 Cofidece Iterval Estimate Proportio Assumptios Two Categorical Outcomes Populatio Follows Biomial Distributio Normal Approximatio Ca Be Used p 5 & (1 - p) 5 Cofidece Iterval Estimate p s ps( 1 Z a / p s p 2 ) p s ps( 1 Z a / 2 p s ) 1999 Pretice-Hall, Ic. Chap. 7-23

24 Example: Estimatig Proportio p s A radom sample of 400 Voters showed 32 preferred Cadidate A. Set up a 95% cofidece iterval estimate for p. ps( 1 Z a / p s p 2. 08( 1. 08) 400 ) p 1999 Pretice-Hall, Ic. Chap p p s ps( 1 Z a / p s ). 08( ) 400

25 Sample Size Too Big: Requires too much resources Too Small: Wo t do the job 1999 Pretice-Hall, Ic. Chap. 7-25

26 Example: Sample Size 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 Z s Error Roud Up 1999 Pretice-Hall, Ic. Chap. 7-26

27 Example: Sample Size for Proportio What sample size is eeded to be withi ± 5 with 90% cofidece? Out of a populatio of 1,000, we radomly selected 100 of which 30 were defective. Z 2 p( 1 error 2 p Roud Up (. 30 )(. 70 ) Pretice-Hall, Ic. Chap. 7-27

28 Example: Sample Size for Mea Usig fpc What sample size is eeded to be 90% cofidet of beig correct withi ± 5? Suppose the populatio size N = N ( N ) ( 500 1) Z where s error Roud Up 1999 Pretice-Hall, Ic. Chap. 7-28

29 Chapter Summary Discussed Cofidece Iterval Estimatio for the Mea (s Kow) Discussed Cofidece Iterval Estimatio for the Mea (s Ukow) Addressed Cofidece Iterval Estimatio for the Proportio Addressed the Situatio of Fiite Populatios Determied Sample Size 1999 Pretice-Hall, Ic. Chap. 7-29