UNIT 8: INTRODUCTION TO INTERVAL ESTIMATION

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1 STATISTICAL METHODS FOR BUSINESS UNIT 8: INTRODUCTION TO INTERVAL ESTIMATION 8..- Itroductio to iterval estimatio 8..- Cofidece itervals. Costructio ad characteristics Cofidece itervals for the mea Cofidece itervals for the proportio Cofidece itervals for the variace Uiversidad de Oviedo. Facultad de Ecoomía y. Grado e ADE. Métodos Estadísticos para la

2 UNIT 8. GOALS Describe advatages ad disadvatages of poit estimates ad iterval estimates. Iterpret the characteristics of precisio ad cofidece i estimatios. Build cofidece itervals for the populatio mea. Determie sample size for the mea. Build cofidece itervals for the populatio variace ad proportio. Determie sample size for the proportio. Uiversidad de Oviedo. Facultad de Ecoomía y. Grado e ADE. Métodos Estadísticos para la

3 Types of estimates GOAL Approximate populatio parameters which are ukow The estimates of the parameters ca be expressed as: A) Poit Estimate Disadvatage: there is o measure of how good the estimate is B) Iterval Estimate Specify a rage withi which the parameter is estimated to lie Margi of error Poit estimate Uiversidad de Oviedo. Facultad de Ecoomía y. Grado e ADE. Métodos Estadísticos para la

4 Relatioship betwee precisio ad cofidece level Estimatio of p Precisio Cofidece $p Maximum Miimum 0% $p $p 0 Miimum Maximum 00% Uiversidad de Oviedo. Facultad de Ecoomía y. Grado e ADE. Métodos Estadísticos para la

5 Buildig Cofidece Itervals POPULATION X Simple radom sample (X, X,...,X ) θ is a ukow parameter Estimator T(X, X,...,X ) P(T θ T ) T T Pivotal quatity d T (X, X,...,X, θ) Cofidece level - P(a d b) T a b Uiversidad de Oviedo. Facultad de Ecoomía y. Grado e ADE. Métodos Estadísticos para la

6 Example. Buildig a cofidece iterval for the populatio mea Simple radom sample (X, X,...,X ) POPULATION X N(µ,) Is kow µ is ukow Estimator X i X i X.96, X +.96 X d X Pivotal quatity X µ N(0,) Cofidece level P(.96 dn ( 0,).96) Uiversidad de Oviedo. Facultad de Ecoomía y. Grado e ADE. Métodos Estadísticos para la

7 Iterval Precisio Iterval precisio is measured through the width of the iterval, A. I case of a symmetric iterval, precisio ca be the evaluated through the margi of error, εa/ WidthAT -T Factors ifluecig precisio T T Margi of error ε Cofidece level Iformatio about the populatio (probability model ad parameters) Sample iformatio (size, samplig methods, estimator) Uiversidad de Oviedo. Facultad de Ecoomía y. Grado e ADE. Métodos Estadísticos para la

8 Iterpretatio of cofidece level (-) If idepedet samples are take repeatedly from the same populatio, ad a cofidece iterval CI calculated for each sample, THEN a (-)% of the itervals will iclude the ukow populatio parameter. Statistics Glossary Parameter CI CI does ot iclude the parameter CI 5 CI does ot iclude the parameter ( CI 4 ) CI 3... Uiversidad de Oviedo. Facultad de Ecoomía y. Grado e ADE. Métodos Estadísticos para la

9 Cofidece iterval for populatio mea NORMAL POPULATION is kow - d X X µ Stadard Normal Distributio N(0,) N(0,) POPULATION IS NOT NORMAL is kow ad is large Obtaig value k P( k N(0,) k) P(T µ T ) T X T P Cofidece iterval for populatio mea with a cofidece level - X k,x + k Uiversidad de Oviedo. Facultad de Ecoomía y. Grado e ADE. Métodos Estadísticos para la

10 Cofidece iterval for populatio mea NORMAL POPULATION is ukow d X µ S t X Studet's t-distributio - P Obtaiig value k ( k t k) T X T P(T µ T ) Cofidece iterval for populatio mea with a cofidece level - X k S,X + k S Uiversidad de Oviedo. Facultad de Ecoomía y. Grado e ADE. Métodos Estadísticos para la

11 CI for µ whe the distributio of the populatio is ukow Tchebysheff's iequality allows to build cofidece itervals whe the distributio of the populatio is ot kow, is kow ad is small P ( d E( d ) < k ) d X X X k k d X X µ E( d ) X 0 Var( d ) X P X < µ < X + Cofidece iterval for populatio mea with a cofidece level - X, X + Uiversidad de Oviedo. Facultad de Ecoomía y. Grado e ADE. Métodos Estadísticos para la

12 Cofidece itervals for populatio mea Situatio Cofidece Iterval Cofidece Level X is Normal, is kow X is ot Normal, is kow ad is large X k,x + k k is foud i N(0,) tables X is Normal, is ukow S X k,x + k S k is foud is Studet s t-distributio (d.f. -) X follows a ukow distributio, is kow ad is small X,X + Uiversidad de Oviedo. Facultad de Ecoomía y. Grado e ADE. Métodos Estadísticos para la

13 GOAL To estimate µ, give a certai cofidece level (- ) ad certai margi of error ε X N( µ, ) Sample size for estimatig µ Sample size X k,x + k ε k k ε X is ot kow X,X + ε ε Sample size icreases with cofidece level Sample size icreases with the value of the stadard deviatio Sample size icreases as the margi of error ε decreases Uiversidad de Oviedo. Facultad de Ecoomía y. Grado e ADE. Métodos Estadísticos para la

14 Cofidece iterval for populatio proportios is large d pˆ pˆ p p( p) Stadard Normal Distributio N(0,) - N(0,) pˆ( pˆ ) Obtaig value k P( k N(0,) k) T $p T P(T p T ) Cofidece iterval for populatio proportio with a cofidece level - pˆ k pˆ( pˆ),pˆ + k pˆ( pˆ) Uiversidad de Oviedo. Facultad de Ecoomía y. Grado e ADE. Métodos Estadísticos para la

15 Goal Sample size for estimatig p To estimate p, give a certai cofidece level (-) ad certai margi of error ε p( p) p( p) pˆ k, pˆ + k Sample size p( p) p( p ε k k ε ) Sice p is ukow, there are to basic optios to approximate p(-p): Carry out a pilot survey i order to a get a prelimiar estimatio of p The upper limit of p(-p) is assumed, p(-p) 0.5 Uiversidad de Oviedo. Facultad de Ecoomía y. Grado e ADE. Métodos Estadísticos para la

16 Cofidece iterval for populatio variace NORMAL POPULATION d S ( )S χ Chi-square distributio Obtaiig values k ad k - P(k χ k) k k P ( χ k ) P( χ k ) Cofidece iterval for populatio variace with a cofidece level - ( )S k ( )S, k Uiversidad de Oviedo. Facultad de Ecoomía y. Grado e ADE. Métodos Estadísticos para la