STATISTICAL METHODS FOR BUSINESS UNIT 8: INTRODUCTION TO INTERVAL ESTIMATION 8.1.- Introduction to interval estimation 8.2.- Confidence intervals. Construction and characteristics 8.3.- Confidence intervals for the mean 8.4.- Confidence intervals for the proportion 8.5.- Confidence intervals for the variance Universidad de Oviedo. Facultad de Economía y Empresa. Grado en ADE. Métodos Estadísticos para la Empresa UNIT 8. GOALS • Describe advantages and disadvantages of point estimates and interval estimates. •Interpret the characteristics of precision and confidence in estimations. • Build confidence intervals for the population mean. • Determine sample size for the mean. • Build confidence intervals for the population variance and proportion. • Determine sample size for the proportion. Universidad de Oviedo. Facultad de Economía y Empresa. Grado en ADE. Métodos Estadísticos para la Empresa Types of estimates GOAL Approximate population parameters which are unknown The estimates of the parameters can be expressed as: A) Point Estimate Disadvantage: there is no measure of how good the estimate is B) Interval Estimate Specify a range within which the parameter is estimated to lie Margin of error Point estimate Universidad de Oviedo. Facultad de Economía y Empresa. Grado en ADE. Métodos Estadísticos para la Empresa Relationship between precision and confidence level Precision Estimation of p Maximum p$ ( p$1 ( 0 Confidence Minimum 0% ) p$2 ) 1 Minimum Maximum 100% Universidad de Oviedo. Facultad de Economía y Empresa. Grado en ADE. Métodos Estadísticos para la Empresa Building Confidence Intervals POPULATION X Simple random sample (X1, X2, ...,Xn) θ is an unknown paramenter P(T1 ≤ θ ≤ T2 ) = 1− α Estimator T(X1, X2, ...,Xn) ( ) T1 T2 P(a ≤ dT ≤ b) = 1 − α Pivotal quantity dT(X1, X2, ...,Xn, θ) Confidence level 1-α ( a ) b Universidad de Oviedo. Facultad de Economía y Empresa. Grado en ADE. Métodos Estadísticos para la Empresa Example. Building a confidence interval for the population mean POPULATION X≈ ≈N(µ µ,σ σ) σ Is known Simple random sample (X1, X2, ...,Xn) Estimator µ is unknown σ σ X − 1.96 n , X + 1.96 n n X= ∑X i =1 ) X n Pivotal quantity dX = ( i X−µ ≈ N(0,1) σ n P(−1.96 ≤ d N ( 0,1) ≤ 1.96) = 0.95 Confidence level 1 - α = 0.95 ( -1.96 ) 1.96 Universidad de Oviedo. Facultad de Economía y Empresa. Grado en ADE. Métodos Estadísticos para la Empresa Interval Precision Interval precision is measured through the width of the interval, A. In case of a symmetric interval, precision can be then evaluated through the margin of error, ε=A/2 WidthA=T2-T1 ( T 1 Factors influencing precision )T Margin of error ε 2 • Confidence level • Information about the population (probability model and parameters) • Sample information (size, sampling methods, estimator) Universidad de Oviedo. Facultad de Economía y Empresa. Grado en ADE. Métodos Estadísticos para la Empresa Interpretation of confidence level (1-α) Parameter “If independent samples are taken repeatedly from the same population, and a confidence interval calculated for each sample, THEN a (1-α)% of the intervals will include the unknown population parameter”. ( ) CI1 ( ) CI2 ( CI3 CI does not include the parameter ( ) CI4 ( ) CI5 Statistics Glossary http://www.stats.gla.ac.uk/ ) ( ) .... CI does not include the parameter Universidad de Oviedo. Facultad de Economía y Empresa. Grado en ADE. Métodos Estadísticos para la Empresa Confidence interval for population mean NORMAL POPULATION σ is known POPULATION IS NOT NORMAL σ is known and n is large dX = X−µ ≈ N(0,1) σ n Standard Normal Distribution N(0,1) 1- α Obtaning value k P( −k ≤ N(0,1) ≤ k ) = 1 − α ( T1 ) X T2 Confidence interval for population mean with a confidence level 1 - α P(T1 ≤ µ ≤ T2 ) = 1− α σ σ ,X + k X − k n n Universidad de Oviedo. Facultad de Economía y Empresa. Grado en ADE. Métodos Estadísticos para la Empresa Confidence interval for population mean dX = NORMAL POPULATION σ is unknown X−µ ≈ t n −1 S n Student's t-distribution 1- α Obtaining value k P ( −k ≤ t n − 1 ≤ k ) = 1 − α ( ) T1 X P(T1 ≤ µ ≤ T2 ) = 1− α T2 Confidence interval for population mean with a confidence level 1 - α S S ,X + k X − k n n Universidad de Oviedo. Facultad de Economía y Empresa. Grado en ADE. Métodos Estadísticos para la Empresa CI for µ when the distribution of the population is unknown Tchebysheff's inequality allows to build confidence intervals when the distribution of the population is not known, σ is known and n is small ( P d X − E(d X ) < kσ d dX = X−µ σ n X ) 1 ≥ 1− 2 = 1− α k E(d X ) = 0 k= 1 α Var (d X ) = 1 1 σ 1 σ ≥ 1 − α P X − <µ<X+ α n α n Confidence interval for population mean with a confidence level 1 - α 1 σ 1 σ , X+ X − α n α n Universidad de Oviedo. Facultad de Economía y Empresa. Grado en ADE. Métodos Estadísticos para la Empresa Confidence intervals for population mean Situation X is Normal, σ is known X is not Normal, σ is known and n is large X is Normal, σ is unknown Confidence Interval σ σ ,X + k X − k n n Confidence Level 1−α k is found in N(0,1) tables S S ,X + k X − k n n 1−α k is found is Student’s t-distribution (d.f.= n-1) X follows an unknown distribution, σ is known and n is small 1 σ 1 σ ,X + X − α n α n ≥1−α Universidad de Oviedo. Facultad de Economía y Empresa. Grado en ADE. Métodos Estadísticos para la Empresa Sample size for estimating µ GOAL To estimate µ, given a certain confidence level (1α) and certain margin of error ε X ≈ N(µ, σ) σ σ ,X + k X − k n n Sample size ε=k σ n kσ n= ε 2 X is not known 1 σ 1 σ ,X + X − α n α n 1 σ ε= α n σ2 n= αε 2 • Sample size increases with confidence level • Sample size increases with the value of the standard deviation • Sample size increases as the margin of error ε decreases Universidad de Oviedo. Facultad de Economía y Empresa. Grado en ADE. Métodos Estadísticos para la Empresa Confidence interval for population proportions dp̂ = n is large p̂ − p p(1 − p) n ≈ N(0,1) p̂(1 − p̂) n−1 Standard Normal Distribution N(0,1) Obtaning value k 1- α P( −k ≤ N(0,1) ≤ k ) = 1 − α ( ) T1 p$ P(T1 ≤ p ≤ T2 ) = 1− α T2 Confidence interval for population proportion with a confidence level 1 - α p̂(1−p̂) p̂(1−p̂) p̂−k ,p̂+k n−1 n−1 Universidad de Oviedo. Facultad de Economía y Empresa. Grado en ADE. Métodos Estadísticos para la Empresa Sample size for estimating p Goal To estimate p, given a certain confidence level (1-α α) and certain margin of error ε p(1 − p) p(1 − p) , pˆ + k pˆ − k n n Sample size p(1 − p) ε=k n n = k2 p(1 − p) ε2 Since p is unknown, there are to basic options to approximate p(1-p): • Carry out a pilot survey in order to a get a preliminar estimation of p • The upper limit of p(1-p) is assumed, p(1-p)= 0.25 Universidad de Oviedo. Facultad de Economía y Empresa. Grado en ADE. Métodos Estadísticos para la Empresa Confidence interval for population variance NORMAL POPULATION dS2 (n − 1)S 2 2 χ = ≈ n −1 σ2 Chi-square distribution Obtaining values k1 and k2 P(k1 ≤ χn2−1 ≤ k2 ) = 1− α 1- α k1 P(χ 2 n −1 α ≤ k1) = 2 k2 P(χ 2 n −1 α ≥ k2 ) = 2 Confidence interval for population variance with a confidence level 1 - α (n−1)S2 (n−1)S2 , k k 2 1 Universidad de Oviedo. Facultad de Economía y Empresa. Grado en ADE. Métodos Estadísticos para la Empresa