Chapter 8 Cofidece Iterval Estimatio Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 1
Learig Objectives I this chapter, you lear: To costruct ad iterpret cofidece iterval estimates for the populatio mea ad the populatio proportio To determie the sample size ecessary to develop a cofidece iterval for the populatio mea or populatio proportio Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 2
Chapter Outlie Cotet of this chapter Cofidece Itervals for the Populatio Mea, μ whe Populatio Stadard Deviatio σ is Kow whe Populatio Stadard Deviatio σ is Ukow Cofidece Itervals for the Populatio Proportio, π Determiig the Required Sample Size Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 3
Poit ad Iterval Estimates A poit estimate is a sigle umber, a cofidece iterval provides additioal iformatio about the variability of the estimate Lower Cofidece Limit Poit Estimate Width of cofidece iterval Upper Cofidece Limit Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 4
Poit Estimates We ca estimate a Populatio Parameter Mea μ with a Sample Statistic (a Poit Estimate) X Proportio π p Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 5
Cofidece Itervals How much ucertaity is associated with a poit estimate of a populatio parameter? A iterval estimate provides more iformatio about a populatio characteristic tha does a poit estimate Such iterval estimates are called cofidece itervals Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 6
Cofidece Iterval Estimate A iterval gives a rage of values: Takes ito cosideratio variatio i sample statistics from sample to sample Based o observatios from 1 sample Gives iformatio about closeess to ukow populatio parameters Stated i terms of level of cofidece e.g. 95% cofidet, 99% cofidet Ca ever be 100% cofidet Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 7
Cofidece Iterval Example Cereal fill example Populatio has µ = 368 ad σ = 15. If you take a sample of size = 25 you kow 368 ± 1.96 * 15 / 25 = (362.12, 373.88). 95% of the itervals formed i this maer will cotai µ. Whe you do t kow µ, you use X to estimate µ If X = 362.3 the iterval is 362.3 ± 1.96 * 15 / 25 = (356.42, 368.18) Sice 356.42 µ 368.18 the iterval based o this sample makes a correct statemet about µ. But what about the itervals from other possible samples of size 25? Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 8
Cofidece Iterval Example (cotiued) Sample # X Lower Limit Upper Limit Cotai µ? 1 362.30 356.42 368.18 Yes 2 369.50 363.62 375.38 Yes 3 360.00 354.12 365.88 No 4 362.12 356.24 368.00 Yes 5 373.88 368.00 379.76 Yes Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 9
Cofidece Iterval Example I practice you oly take oe sample of size I practice you do ot kow µ so you do ot kow if the iterval actually cotais µ However you do kow that 95% of the itervals formed i this maer will cotai µ Thus, based o the oe sample, you actually selected you ca be 95% cofidet your iterval will cotai µ (this is a 95% cofidece iterval) Note: 95% cofidece is based o the fact that we used Z = 1.96. (cotiued) Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 10
Estimatio Process Populatio (mea, μ, is ukow) Radom Sample Mea X = 50 I am 95% cofidet that μ is betwee 40 & 60. Sample Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 11
Geeral Formula The geeral formula for all cofidece itervals is: Poit Estimate ± (Critical Value)(Stadard Error) Where: Poit Estimate is the sample statistic estimatig the populatio parameter of iterest Critical Value is a table value based o the samplig distributio of the poit estimate ad the desired cofidece level Stadard Error is the stadard deviatio of the poit estimate Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 12
Cofidece Level Cofidece the iterval will cotai the ukow populatio parameter A percetage (less tha 100%) Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 13
Cofidece Level, (1-a) (cotiued) Suppose cofidece level = 95% Also writte (1 - a) = 0.95, (so a= 0.05) A relative frequecy iterpretatio: 95% of all the cofidece itervals that ca be costructed will cotai the ukow true parameter A specific iterval either will cotai or will ot cotai the true parameter No probability ivolved i a specific iterval Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 14
Cofidece Itervals Cofidece Itervals Populatio Mea Populatio Proportio σ Kow σ Ukow Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 15
8.1.Cofidece Iterval for the mea μ (σ Kow) Assumptios Populatio stadard deviatio σ is kow Populatio is ormally distributed If populatio is ot ormal, use large sample ( > 30) Cofidece iterval estimate: X ± Zα/2 σ where σ/ X Z α/2 is the poit estimate is the ormal distributio critical value for a probability of a/2 i each tail is the stadard error Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 16
Fidig the Critical Value, Z α/2 Cosider a 95% cofidece iterval: 1 α = 0.95 so α = 0.05 Z α/2 = ± 1.96 α 2 = 0.025 α 2 = 0.025 Z uits: X uits: Z α/2 = -1.96 Z α/2 = 1.96 Lower Cofidece Limit 0 Poit Estimate Upper Cofidece Limit Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 17
Commo Levels of Cofidece Commoly used cofidece levels are 90%, 95%, ad 99% Cofidece Level 80% 90% 95% 98% 99% 99.8% 99.9% Cofidece Coefficiet, 1 α 0.80 0.90 0.95 0.98 0.99 0.998 0.999 Z α/2 value 1.28 1.645 1.96 2.33 2.58 3.08 3.27 Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 18
Itervals ad Level of Cofidece Samplig Distributio of the Mea Itervals exted from X Z α / to X + Z α / 2 2 σ σ α /2 1 α α/2 μ x = μ x 1 x 2 Cofidece Itervals x (1-a)100% of itervals costructed cotai μ; (a)100% do ot. Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 19
Example A sample of 11 circuits from a large ormal populatio has a mea resistace of 2.20 ohms. We kow from past testig that the populatio stadard deviatio is 0.35 ohms. Determie a 95% cofidece iterval for the true mea resistace of the populatio. Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 20
Example A sample of 11 circuits from a large ormal populatio has a mea resistace of 2.20 ohms. We kow from past testig that the populatio stadard deviatio is 0.35 ohms. (cotiued) Solutio: X ± Zα/2 σ = 2.20 ± 1.96 (0.35/ 11) = 2.20 ± 0.2068 1.9932 μ 2.4068 Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 21
Iterpretatio We are 95% cofidet that the true mea resistace is betwee 1.9932 ad 2.4068 ohms Although the true mea may or may ot be i this iterval, 95% of itervals formed i this maer will cotai the true mea Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 22
Cofidece Itervals Cofidece Itervals Populatio Mea Populatio Proportio σ Kow σ Ukow Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 23
Do You Ever Truly Kow σ? Probably ot! I virtually all real world busiess situatios, σ is ot kow. If there is a situatio where σ is kow the µ is also kow (sice to calculate σ you eed to kow µ.) If you truly kow µ there would be o eed to gather a sample to estimate it. Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 24
8.2. Cofidece Iterval for μ (σ Ukow) If the populatio stadard deviatio σ is ukow, we ca substitute the sample stadard deviatio, S This itroduces extra ucertaity, sice S is variable from sample to sample So we use the t distributio istead of the ormal distributio Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 25
Cofidece Iterval for μ (σ Ukow) Assumptios (cotiued) Populatio stadard deviatio is ukow Populatio is ormally distributed If populatio is ot ormal, use large sample ( > 30) Use Studet s t Distributio Cofidece Iterval Estimate: X ± tα / 2 (where t α/2 is the critical value of the t distributio with -1 degrees of freedom ad a area of α/2 i each tail) S Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 26
Studet s t Distributio The t is a family of distributios The t α/2 value depeds o degrees of freedom (d.f.) Number of observatios that are free to vary after sample mea has bee calculated d.f. = - 1 Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 27
Degrees of Freedom (df) Idea: Number of observatios that are free to vary after sample mea has bee calculated Example: Suppose the mea of 3 umbers is 8.0 Let X 1 = 7 Let X 2 = 8 What is X 3? If the mea of these three values is 8.0, the X 3 must be 9 (i.e., X 3 is ot free to vary) Here, = 3, so degrees of freedom = 1 = 3 1 = 2 (2 values ca be ay umbers, but the third is ot free to vary for a give mea) Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 28
Studet s t Distributio Note: t Z as icreases Stadard Normal (t with df = ) t-distributios are bellshaped ad symmetric, but have fatter tails tha the ormal t (df = 13) t (df = 5) 0 t Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 29
Studet s t Table Upper Tail Area df.10.05.025 1 3.078 6.314 12.706 Let: = 3 df = - 1 = 2 a= 0.10 a/2 = 0.05 2 1.886 2.920 4.303 3 1.638 2.353 3.182 a/2 = 0.05 The body of the table cotais t values, ot probabilities 0 2.920 t Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 30
Selected t distributio values With compariso to the Z value Cofidece t t t Z Level (10 d.f.) (20 d.f.) (30 d.f.) ( d.f.) 0.80 1.372 1.325 1.310 1.28 0.90 1.812 1.725 1.697 1.645 0.95 2.228 2.086 2.042 1.96 0.99 3.169 2.845 2.750 2.58 Note: t Z as icreases Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 31
Example of t distributio cofidece iterval A radom sample of = 25 has X = 50 ad S = 8. Form a 95% cofidece iterval for μ d.f. = 1 = 24, so t α /2 = t 0.025 = 2.0639 The cofidece iterval is S X ± t α /2 = 50 ± (2.0639) 8 25 46.698 µ 53.302 Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 32
Example of t distributio cofidece iterval (cotiued) Iterpretig this iterval requires the assumptio that the populatio you are samplig from is approximately a ormal distributio (especially sice is oly 25). This coditio ca be checked by creatig a: Normal probability plot or Boxplot Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 33
Cofidece Itervals Cofidece Itervals Populatio Mea Populatio Proportio σ Kow σ Ukow Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 34
8.3.Cofidece Itervals for the Populatio Proportio, π A iterval estimate for the populatio proportio ( π ) ca be calculated by addig a allowace for ucertaity to the sample proportio ( p ) Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 35
Cofidece Itervals for the Populatio Proportio, π (cotiued) Recall that the distributio of the sample proportio is approximately ormal if the sample size is large, with stadard deviatio σ p = π(1 π ) We will estimate this with sample data: p(1 p) Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 36
Cofidece Iterval Edpoits Upper ad lower cofidece limits for the populatio proportio are calculated with the formula where p ± Z α /2 p(1 Z α/2 is the stadard ormal value for the level of cofidece desired p is the sample proportio is the sample size Note: must have p > 5 ad (1-p) > 5 p) Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 37
Example A radom sample of 100 people shows that 25 are left-haded. Form a 95% cofidece iterval for the true proportio of left-haders Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 38
Example A radom sample of 100 people shows that 25 are left-haded. Form a 95% cofidece iterval for the true proportio of left-haders. (cotiued) p ± Z α /2 p(1 p)/ = 25/100 ± 1.96 0.25(0.75)/100 = 0.25 ± 1.96(0.0433) = 0.1651 π 0.3349 Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 39
Iterpretatio We are 95% cofidet that the true percetage of left-haders i the populatio is betwee 16.51% ad 33.49%. Although the iterval from 0.1651 to 0.3349 may or may ot cotai the true proportio, 95% of itervals formed from samples of size 100 i this maer will cotai the true proportio. Copyright 2015, 2012, 2009 Pearso Educatio, Ic. Chapter 8, Slide 40