Introduction to Probability and Statistics Chapter 7

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1 Itroductio to Probability ad Statistics Chapter 7 Ammar M. Sarha, asarha@mathstat.dal.ca Departmet of Mathematics ad Statistics, Dalhousie Uiversity Fall Semester 008

2 Chapter 7 Statistical Itervals Based o a Sigle Sample Dr. Ammar Sarha

3 Cofidece Itervals A alterative to reportig a sigle value for the parameter beig estimated is to calculate ad report a etire iterval of plausible values a cofidece iterval (CI). A cofidece level is a measure of the degree of reliability of the iterval. Dr. Ammar Sarha 3

4 7.1 Basic Properties of Cofidece Itervals If after observig 1 x 1,, x, we compute the observed sample mea x, the a 95% cofidece iterval for μ ca be expressed as σ σ x 1.96 x 1.96 The populatio has a N(μ, σ ) ad σ is kow., Dr. Ammar Sarha 4

5 Other Levels of Cofidece A (1- α)100% cofidece iterval for the mea μ of a ormal populatio whe the value of is kow α is give by x σ x, σ Other Levels of Cofidece Dr. Ammar Sarha 5

6 Sample Sie The geeral formula for the sample sie ecessary to esure a iterval width w is σ w Dr. Ammar Sarha 6

7 Derivig a Cofidece Iterval Let 1,, deote the sample o which the CI for the parameter θ is to be based. Suppose a radom variable satisfyig the followig properties ca be foud: 1. The variable depeds fuctioally o both 1,, ad θ.. The probability distributio of the variable does ot deped o θ or ay other ukow parameters. Let h( 1,, ; θ) deote this radom variable. I geeral, the form of h is usually suggested by examiig the distributio of a appropriate estimator θˆ. For ay α, 0 < α < 1, costats a ad b ca be foud to satisfy P ( a < h ( 1, L, ; θ ) < b ) 1 α Dr. Ammar Sarha 7

8 Now suppose that the iequalities ca be maipulated to isolate θ P ( l, L, ) < θ < u (, L, )) 1 α ( 1 1 lower cofidece limit upper cofidece limit For a 100(1- α)% CI. Dr. Ammar Sarha 8

9 Examples: 7.5 p. 60. will be give i the class. Dr. Ammar Sarha 9

10 7. Large-Sample Cofidece Itervals for a Populatio Mea ad Proportio Large-Sample Cofidece Iterval Let 1,, deote the sample from a populatio havig a mea μ ad stadard deviatio σ. If is sufficietly large, the This implies that σ σ (0,1) is a large-sample cofidece iterval for μ with level 100(1- α)%. Z, This formula is valid regardless of the shape of the populatio distributio. For practice: > 40. ( μ, ) σ ~ N σ μ ~ N Dr. Ammar Sarha 10

11 Notice, if σ is ukow, replace it with the sample stadard deviatio s. That is, Examples: 7.6 p. 64. Suppose a radom sample with sie 48 from a populatio with ukow mea μ ad ukow variace σ with the followig iformatio: s, x i 66, ad x x 54.7 ad s 5.3 i , ,950 Fid the 95% cofidece iterval of μ? Solutio: From the iformatio give i the problem, we have: The, the 95% cofidece iterval of μ is ( 53., 56. ) That is, with a cofidece level of approximatio 95%, 53. < μ < 56. s Dr. Ammar Sarha 11

12 Dr. Ammar Sarha 1 Cofidece Iterval for a Populatio Proportio Let p deote the proportio of successes i a populatio, where success idetifies a idividual or object that has a specified property. A radom sample of idividuals is to be selected, ad is the umber of successes i the sample. A cofidece iterval for a populatio proportio p with level 100(1- α)% is: ( ) ( ) q p p q p p 1 4 ˆ ˆ ˆ, 1 4 ˆ ˆ ˆ α α α α α α α α where,. ˆ 1 ˆ, ˆ p q p

13 Notice, if the sample sie is quite large, the approximate CI limits become pˆ qˆ pˆ pˆ, Sice α () is egligible compared to pˆ. pˆ qˆ Sample Sie The geeral formula for the sample sie ecessary to esure a iterval width w is 4 pˆ qˆ w Dr. Ammar Sarha 13

14 Examples: 7.8 p. 67. Suppose that i 48 trials i a particular laboratory, 16 resulted i igitio of a particular type of substrate by a lighted cigarette. Fid the 95% cofidece iterval of the log-ru proportio of all such trails that would result i igitio? Solutio: 48 Let p deote the log-ru proportio of all such trails that would result i igitio pˆ 0.333, q 48 3 The, the approximately 95% CI of p is (1.96 ) ( 48 ) ± ± (0.333 )( ) 48 ( 1.96 ) 48 (0.17, 0.474) (1.96 ) 4 ( 48 ) Dr. Ammar Sarha 14

15 Notice, the traditioal 95%CI is (0.333 )( ) ± (0.00, ) Examples: 7.9 p. 67. Fid the sample sie ecessary to esure a width of 0.10 for the 95% cofidece iterval of the log-ru proportio of all such trails that would result i igitio? 4 pˆ qˆ w 4*(1.96) * 0.333* Dr. Ammar Sarha 15

16 7.3 Itervals Based o a Normal Populatio Distributio The populatio of iterest is ormal, so that 1,, costitutes a radom sample from a ormal distributio with both μ ad σ ukow. t Distributio Let 1,, ~ N(μ, σ ), the the rv T S μ has a probability distributio called a t distributio with -1 degrees of freedom (df). Dr. Ammar Sarha 16

17 Properties of t Distributios Let t v deote the desity fuctio curve for v df. 1. Each t v curve is bell-shaped ad cetered at 0.. Each t v curve is spread out more tha the stadard ormal () curve. 3. As v icreases, the spread of the correspodig t v curve decreases. 4. As v, the sequece of t v curves approaches the stadard ormal curve (the curve is called a t curve with df ) curve t 5 curve t 5 curve 0 Dr. Ammar Sarha 17

18 t Critical Value Let t α,v the umber o the measuremet axis for which the area uder the t curve with v df to the right of t α,v is α. t α,v is called t critical value. t critical value Table A.5, p. 671, gives the t critical value for give α, v. t 0.05,15.131, t 0.05, 1.717, t 0.01,.508. Dr. Ammar Sarha 18

19 t Cofidece Iterval Now, let 1,, be a radom sample from a ormal distributio with both μ ad σ ukow, the the (1- α)100% CI of μ is or t S, 1,, 1 t ±, 1 Here, ad S are the sample mea ad sample variace. s t S Dr. Ammar Sarha 19

20 Examples: 7.1 p. 74. Cosider the followig sample of fat cotet (i percetage) o 10 radomly selected hot dogs: Fid the 95% cofidece iterval of the populatio mea fat cotet, assumig the populatio is ormal.? Solutio: We have: 10, x 1.90 ad s 4.134, The, the 95% cofidece iterval of μ is t 0.05,10 1, 1.90 t 0.05, ( 18.94, 4.86 ) , Dr. Ammar Sarha 0

21 Summary The radom sample 1,, ~ N(μ, σ ) σ is kow (1- α)100% CI of μ ± σ 1,, ~ N(μ, σ ) σ is ukow t ±, 1 S 1,, ~ ay populatio with mea μ ad variace σ σ is kow ( 30) ± σ σ is ukow ( 30) ± S Dr. Ammar Sarha 1

point estimator a random variable (like P or X) whose values are used to estimate a population parameter

point estimator a random variable (like P or X) whose values are used to estimate a population parameter Estimatio We have oted that the pollig problem which attempts to estimate the proportio p of Successes i some populatio ad the measuremet problem which attempts to estimate the mea value µ of some quatity