Walpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 8-1Chapter 8Sampling istributionWalpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 8-!Chapter "oalsAfter completing this chapter, you should be able to: efine the concept of a sampling distributionetermine the mean and standard de#iation for the sampling distribution of the sample mean, $Sampling distribution of meansCentral %imit &heoremSampling distribution of S!Sampling distribution of ps ''Walpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 8-(Sampling istributions) sampling distribution is a distribution of all of the possible #alues of a statistic for a gi#en si*e sample selected from a populationWalpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 8-+Sampling istributionAssume there is a population Population si*e ,-+.andom #ariable, $,is age of indi#iduals/alues of $0 18, !1,!!, !+ 2years3ABCDWalpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 8-4.3.2.1 1! 222 2" ABC D5niform istributionP2636(continued)Summary 7easures for the Population istribution0Sampling istribution!1+!+ !! !1 18,$8i=+ + +== !.!(9,83 2$:!i==Walpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 8-919 possible samples 2sampling ;ith replacement3,o; consider all possible samples of si*e n-!1st2nd #bser$ation Obs1!2222" 1!181f the Population is ,ormal>f a population is normal ;ith mean 8 and standard de#iation :, the sampling distribution ofis also normally distributed ;ithand2&his assumes that sampling is ;ith replacement or sampling is ;ithout replacement from an infinite population3$8 8$ =n::$ =Walpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 8-1+?-#alue for Sampling istributionof the 7ean?-#alue for the sampling distribution of 0;here0 - sample mean- population mean- population standard de#iation n - sample si*e$8:n:83 $ 2:3 8 $ 2?$$==$Walpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 8-14@inite Population Correction)pply the *inite %opulation Correction if0the sample is large relati#e to the population 2nis greater than 4A of,3andBSampling is ;ithout replacement&hen1 ,n ,n:83 $ 2?=Walpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 8-19,ormal Population istribution,ormal Sampling istribution 2has the same mean3Sampling istribution Properties

2i.e.is unbiased 36668 86 =868Walpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 8-1=Sampling istribution Properties@or sampling ;ith replacement0 )s nincreases,decreases+arger sample si,e-maller sample si,e6(continued)6:8Walpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 8-18>f the Population is not ,ormalWe can apply the Central %imit &heorem0E#en if the population is not normal,Bsample means from the population ;ill be appro6imately normal as long as the sample si*e is large enough.Properties of the sampling distribution0 and8 86 =n::6 =Walpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 8-1f the Population is not ,ormal(continued)Sampling distribution properties08 86 =n::6 =688Walpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 8-!1Do; %arge is %arge EnoughE@or most distributions, n F (1;ill gi#e a sampling distribution that is nearly normal@or fairly symmetric distributions, n F 14@or normal population distributions, the sampling distribution of the mean is al;ays normally distributedWalpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 8-!!E6ampleSuppose a population has mean 8 - 8 and standard de#iation : - (.Suppose a random sample of si*e n - (9 is selected.What is the probability that the sample mean is bet;een =.8 and 8.!EWalpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 8-!(E6ampleSolution0E#en if the population is not normally distributed, the central limit theorem can be used 2n F (13B so the sampling distribution ofis appro6imately normalB ;ith mean-8 Band standard de#iation (continued)6681.4(9(n::6= = =Walpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 8-!+E6ample Solution 2continued30(continued)0.3830 0.4) Z P(-0.43638 - 8.2n - 3638 - 7.8P 8.2) P(7.8X= < < =< < = < dea0 ,umber of obser#ations that are free to #ary after sample mean has been calculated45ample: Suppose the mean of ( numbers is 8.1 %et $1 - =%et $! - 8What is '3EWalpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 8-(9t-istributiont1t2df - 43 t2df - 1(3t-distributions are bell-shaped and symmetric, but ha#e NfatterM tails than the normalStandard ,ormal2t ;ith df - 3,ote0t ?asnincreasesWalpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 8-(=t &able6pper 7ail Areadf .!4 .11.21 1.111 (.1=8 9.(1+21.81= 1.889 2.82( 1.=94 1.9(8 !.(4(t

2.82&he body of the table contains t #alues, not probabilities%et0 n - ( df - n - 1 - ! - .11 O! -.14O! - .14Walpole,Probability and Statistics for Engineers & Scientists, 8th e., Pearson Edu. Chap 8-(8t distribution #aluesWith comparison to the ? #alueConfidence t tt9+e$el&1 d.f.( &2 d.f.( &3 d.f.( )))) .81 1.(=!1.(!4 1.(11 1.!8 .