7-1. Chapter 4. Part I. Sampling Distributions and Confidence Intervals

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1 7-1 Chapter 4 Part I. Samplig Distributios ad Cofidece Itervals 1

2 7- Sectio 1. Samplig Distributio

3 7-3 Usig Statistics Statistical Iferece: Predict ad forecast values of populatio parameters... Test hypotheses about values of populatio parameters... Make decisios... O basis of sample statistics derived from limited ad icomplete sample iformatio Make geeralizatios about the characteristics of a populatio... O the basis of observatios of a sample, a part of a populatio 3

4 Sample Statistics as Estimators of Populatio Parameters 7-4 A sample statistic is a umerical measure of a summary characteristic of a sample. A populatio parameter is a umerical measure of a summary characteristic of a populatio. A estimator of a populatio parameter is a sample statistic used to estimate or predict the populatio parameter. A estimate of a parameter is a particular umerical value of a sample statistic obtaied through samplig. A poit estimate is a sigle value used as a estimate of a populatio parameter. 4

5 7-5 Estimators The sample mea,, is the most commo estimator of the populatio mea, The sample variace, s, is the most commo estimator of the populatio variace,. The sample stadard deviatio, s, is the most commo estimator of the populatio stadard deviatio,. The sample proportio, pˆ, is the most commo estimator of the populatio proportio, p. 5

6 A Populatio Distributio, a Sample from a Populatio, ad the Populatio ad Sample Meas 7-6 Populatio mea () Frequecy distributio of the populatio Sample poits Sample mea ( ) 6

7 7-7 Other Samplig Methods Stratified samplig: i stratified samplig, the populatio is partitioed ito two or more subpopulatio called strata, ad from each stratum a desired sample size is selected at radom. Cluster samplig: i cluster samplig, a radom sample of the strata is selected ad the samples from these selected strata are obtaied. Systemic samplig: i systemic samplig, we start at a radom poit i the samplig frame, ad from this poit selected every k th, say, value i the frame to formulate the sample. 7

8 7-8 Samplig Distributios The samplig distributio of a statistic is the probability distributio of all possible values the statistic may assume, whe computed from radom samples of the same size, draw from a specified populatio. The samplig distributio of is the probability distributio of all possible values the radom variable may assume whe a sample of size is take from a specified populatio. 8

9 Properties of the Samplig Distributio of the Sample Mea 7-9 Comparig the populatio distributio ad the samplig distributio of the mea: The samplig distributio is more bell-shaped ad symmetric. Both have the same ceter. The samplig distributio of the mea is more compact, with a smaller variace. P() Uiform Distributio (1,8) E( ) Var( ) P() Samplig Distributio of the Mea

10 7-10 Samplig from a Normal Populatio Whe samplig from a ormal populatio with mea ad stadard deviatio, the sample mea,, has a ormal samplig distributio: ~ N(, ) This meas that, as the sample size icreases, the samplig distributio of the sample mea remais cetered o the populatio mea, but becomes more compactly distributed aroud that populatio mea f() Samplig Distributio of the Sample Mea Samplig Distributio: =16 Samplig Distributio: = 4 Samplig Distributio: = Normal populatio 10

11 7-11 The Cetral Limit Theorem Whe samplig from a populatio with mea ad fiite stadard deviatio, the samplig distributio of the sample mea will ted to a ormal distributio with mea ad stadard deviatio as the sample size becomes large ( >30). P() = 5 = 0 P() 0.1 For large eough : ~ N (, / ) 0.0 Large f()

12 Studet s t Distributio 7-1 If the populatio stadard deviatio,, is ukow, replace with the sample stadard deviatio, s. If the populatio is ormal, the resultig statistic: t s / has a t distributio with ( - 1) degrees of freedom. The t is a family of bell-shaped ad symmetric distributios, oe for each umber of degree of freedom. The epected value of t is 0. The t distributio approaches a stadard ormal as the umber of degrees of freedom icreases. Stadard ormal t, df=0 t, df=10

13 The Samplig Distributio of the Sample Proportio, p 7-13 The sample proportio is the percetage of successes i biomial trials. It is the umber of successes,, divided by the umber of trials,. P() =, p = 0.3 Sample proportio: p P() =10,p=0.3 As the sample size,, icreases, the samplig distributio of p approaches a ormal distributio with mea p ad stadard deviatio p( 1 p) P() =15, p = ^p 13

14 Estimators ad Their Properties 7-14 A estimator of a populatio parameter is a sample statistic used to estimate the parameter. The most commoly-used estimator of the: Populatio Parameter Sample Statistic Mea () is the Mea ( ) Variace ( ) is the Variace (s ) Stadard Deviatio () is the Stadard Deviatio (s) Proportio (p) is the Proportio ( p ) Desirable properties of estimators iclude: Ubiasedess Efficiecy Cosistecy Sufficiecy 14

15 Types of Estimators 7-15 Poit Estimate A sigle-valued estimate. A sigle elemet chose from a samplig distributio. Coveys little iformatio about the actual value of the populatio parameter, about the accuracy of the estimate. Cofidece Iterval or Iterval Estimate A iterval or rage of values believed to iclude the ukow populatio parameter. Associated with the iterval is a measure of the cofidece we have that the iterval does ideed cotai the parameter of iterest. 15

16 7-16 Sectio. Cofidece Itervals for Populatio Meas (Z-CI ad t-ci) 16

17 Cofidece Iterval for whe is kow 7-17 If the populatio distributio is ormal, the samplig distributio of the mea is ormal. If the sample is sufficietly large ( 30), regardless of the shape of the populatio distributio, the samplig distributio is ormal (Cetral Limit Theorem). I either case : Stadard Normal Distributio 95% : Iterval P or f(z) P z

18 Cofidece Iterval for whe is kow 7-18 Before samplig, there is a 0.95probability that the iterval 1.96 will iclude the sample mea (ad 5% that it will ot). Coversely, after samplig, approimately 95% of such itervals 1.96 will iclude the populatio mea (ad 5% of them will ot). That is, 1.96 is a 95% cofidece iterval for. 18

19 7-19 A 95% Iterval aroud the Populatio Mea 0.4 Samplig Distributio of the Mea f() % fall below the iterval.5% %.5% 196. Approimately 95% of sample meas ca be epected to fall withi the iterval. 196., 196. So 5% ca be epected to fall outside the iterval. 196., % fall above the iterval 95% fall withi the iterval 19

20 95% Itervals aroud the Sample Mea Samplig Distributio of the Mea 95% Approimately 95% of the itervals 1.96 f() % % 196. aroud the sample mea ca be epected to iclude the actual value of the populatio mea,. (Whe the sample mea falls withi the 95% iterval aroud the populatio mea.) * * 0

21 The 95% Cofidece Iterval for 7-1 A 95% cofidece iterval for μ whe σ is kow ad samplig is doe from a ormal populatio, or a large sample is used: 1.96 The quatity 1.96 is ofte called the margi of error or the samplig error. For eample, if: = 5 = 0 = 1 A 95% cofidece iterval: (. 196)( 4) ,

22 A (1- )100% Cofidece Iterval for 7- We defie z as the z value that cuts off a right-tail area of uder the stadard ormal curve. (1-) is called the cofidece coefficiet. is called the error probability, ad (1-)100% is called the cofidece level. f(z) Stadard Normal Distributio z 0 Z 1 z ( 1 ) Pz z Pz z Pz z z ( 1 ) (1- )100% Cofidece Iterval: z

23 Critical Values of z ad Levels of Cofidece 7-3 ( 1 ) z f(z) Stadard Normal Distributio z 0 Z 1 z ( 1 )

24 Level of cofidece ad width of the cofidece iterval 7-4 Whe samplig from the same populatio, usig a fied sample size, the higher the cofidece level, the wider the cofidece iterval. Stadard Normal Distributio Stadard Normal Distributio f(z) 0. f(z) % Cofidece Iterval: Z % Cofidece Iterval: Z

25 7-5 The Sample Size ad the Width of the Cofidece Iterval Whe samplig from the same populatio, usig a fied cofidece level, the larger the sample size,, the arrower the cofidece iterval. S am p lig D istrib utio of the Me a S am p lig D istrib utio of the Me a f() 0. f( ) % Cofidece Iterval: = 0 95% Cofidece Iterval: = 40 Note: The width of a cofidece iterval ca be reduced oly at the price of: a lower level of cofidece, or a larger sample. 5

26 Eample Populatio cosists of the Fortue 500 Compaies (Fortue Web Site), as raked by Reveues. You are tryig to to fid out the average Reveues for the compaies o the list. The populatio stadard deviatio is $15, A radom sample of 30 compaies obtais a sample mea of $10, Give a 95% ad 90% cofidece iterval for the average Reveues 6

27 Chi-square Distributio 7-7 The radom sample is the sample mea ad where 1,, S,, is from a ormal distributio N(, ), is the sample variace, the ( 1) S ~ 1 1 is the chi - square distributio with degrees of freedom ( -1). Property of Chi - square distributiow ~ 1. E(W) r, Var(W) r. If Z ~ N(0,1), the Z 3. Additive Property : the W 1 W ~ 1. Wm ~ r r m r Idepedet Chi - square r.v. W 1 Gamma( r/, i ~ r i 1/ ) : 7

28 t distributio 7-8 r Assume Z~N(0, 1), W ~, Z ad W are idepedet, the Z W / r ~ t r The statistic T ~ t 1 degrees of freedom=(-1) S / Stadard Normal Bell-Shaped Symmetric Fatter Tails t (df = 13) t (df = 5) 0 Z t 8

29 Studet s t Table 7-9 Upper Tail Area r Let: = 3 df = -1 = =.10 / = / = Fid t values: t Values 0.90 t 1. α=0.10, =0. α=0.01, =8 3. α=0.05, =10 9

30 Cofidece itervals for whe is ukow (t distributio) 7-30 A (1-)100% cofidece iterval for whe is ot kow (assumig a ormally distributed populatio): t 1 t 1 where is the value of the t distributio with -1 degrees of freedom that cuts off a tail area of to its right. Eample : A stock market aalyst wats to estimate the average retur o a certai stock. A radom sample of 15 days yields a average (aualized) retur of 10.37% ad a stadard deviatio of s = 3.5%. Assumig a ormal populatio of returs, give a 95% cofidece iterval for the average retur o this stock. s 30

31 7-31 Sectio 3. Cofidece Iterval for Proportios 31

32 Large-Sample Cofidece Itervals for the Populatio Proportio, p 7-3 A large-sample (1-)100% cofidece iterval for the populatio proportio, p: ˆ ˆ pˆ z pq α where the sample proportio, pˆ, is equal to the umber of successes i the sample, divided by the umber of trials (the sample size),, ad qˆ =1-pˆ. For estimatig p, a sample is cosidered large eough whe p 5 ad (1-p) 5, 3

33 Eample A marketig research firm wats to estimate the share that foreig compaies have i the America market for certai products. A radom sample of 100 cosumers is obtaied, ad it is foud that 34 people i the sample are users of foreig-made products; the rest are users of domestic products. Give a 95% cofidece iterval for the share of foreig products i this market. p pq 034)(. 066) z ( 196. )( ) , Thus, the firm may be 95% cofidet that foreig maufacturers cotrol aywhere from 4.7% to 43.8% of the market. 33

34 Cofidece Itervals for the Populatio Variace: The Chi-Square ( ) Distributio 7-34 The sample variace, s, is a ubiased estimator of the populatio variace,. Cofidece itervals for the populatio variace are based o the chi-square ( (r)) distributio. The chi-square distributio is the probability distributio of the sum of several idepedet, squared stadard ormal radom variables. The mea of the chi-square distributio is equal to the degrees of freedom parameter, (E[ ] = r). The variace of a chi-square is equal to twice the umber of degrees of freedom, (Var[ ] = r). 34

35 7-35 The Chi-Square ( ) Distributio The chi-square radom variable caot be egative. The chi-square distributio is skewed to the right. The chi-square distributio approaches a ormal as the degrees of freedom icrease. f ( ) Chi-Square Distributio: df=10, df=30, df=50 I samplig from a ormal populatio, the radom variable: ( ) s 1 has a chi - square distributio with ( -1) degrees of freedom. 0 df = 10 df = df =

36 7-36 Cofidece Iterval for the Populatio Variace A (1-)100% cofidece iterval for the populatio variace * (where the populatio is assumed ormal) is: ( ) s 1, ( 1 ) s 1 where is the value of the chi-square distributio with - 1 degrees of freedom that cuts off a area to its right ad is the value of the distributio that 1 cuts off a area of to its left (equivaletly, a area of 1 to its right). * Note: Because the chi-square distributio is skewed, the cofidece iterval for the populatio variace is ot symmetric 36

37 Cofidece Iterval for the Populatio Variace Eample I a automated process, a machie fills cas of coffee. If the average amout filled is differet from what it should be, the machie may be adjusted to correct the mea. If the variace of the fillig process is too high, however, the machie is out of cotrol ad eeds to be repaired. Therefore, from time to time regular checks of the variace of the fillig process are made. This is doe by radomly samplig filled cas, measurig their amouts, ad computig the sample variace. A radom sample of 30 cas gives a estimate s = 18,540. Give a 95% cofidece iterval for the populatio variace,. ( ), ( ) 1 s 1 s ( )18540, ( ) ,

38 Sample-Size Determiatio 7-38 Before determiig the ecessary sample size, three questios must be aswered: How close do you wat your sample estimate to be to the ukow parameter? (What is the desired boud, B?) What do you wat the desired cofidece level (1-) to be so that the distace betwee your estimate ad the parameter is less tha or equal to B? What is your estimate of the variace (or stadard deviatio) of the populatio i questio? For eample: A (1- ) Cofidece Iterval for : z Boud, B 38

39 Sample Size ad Stadard Error 7-39 The sample size determies the boud of a statistic, sice the stadard error of a statistic shriks as the sample size icreases: Sample size = Stadard error of statistic Sample size = Stadard error of statistic 39

40 Miimum Sample Size: Mea ad Proportio 7-40 Miimum required sample size i estimatig the populatio mea, : z B Boud of estimate: B = z Miimum required sample size i estimatig the populatio proportio, p z pq B 40

41 Sample-Size Determiatio: Eample A marketig research firm wats to coduct a survey to estimate the average amout spet o etertaimet by each perso visitig a popular resort. The people who pla the survey would like to determie the average amout spet by all people visitig the resort to withi $10, with 95% cofidece. From past operatio of the resort, a estimate of the populatio stadard deviatio is s = $400. What is the miimum required sample size? z B (. 196 )( 400 )

42 Sample-Size for Proportio: Eample The maufacturers of a sports car wat to estimate the proportio of people i a give icome bracket who are iterested i the model. The compay wats to kow the populatio proportio, p, to withi 0.01 with 99% cofidece. Curret compay records idicate that the proportio p may be aroud 0.5. What is the miimum required sample size for this survey? z pq B (. )(. )

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