Fundamental Sampling DistributionsLinear Combinations of R.V.’sSlide 3Slide 4SamplingSlide 6EstimatorsSampling DistributionsSampling Distributions - SetupSampling Distributions - MeanSlide 11Slide 12Central Limit Theorem (CLT)Sampling Distributions - MeansSlide 15Slide 16Slide 17Sampling Distributions - VarianceSlide 19Slide 20Slide 21Student’s t-DistributionSlide 23Slide 24F-DistributionSlide 26Slide 27Slide 28Upper Tail Probabilities in ExcelUpper Tail Percentiles in Excel1Fundamental Sampling DistributionsLinear Combinations of R.V.’sSamplingEstimators•Central tendency•Variability or dispersionSampling Distributions•Sample mean and sample variance•Student’s t-distribution•F-distribution2Linear Combinations of R.V.’sX1, X2, … , Xn are independent and normally distributed r.v.’s with means 1, 2, … , n and standard deviations 1, 2, … , n , respectivelyLet Y = a1X1 + a2X2 + … + anXn•Y is normally distributed• • nnaaaYE 2211][2222222121)(nnaaaYV 3Linear Combinations of R.V.’sX1, X2, … , Xn are mutually independent r.v.’s distributed chi-squared with v1, v2, … , vn degrees of freedom , respectivelyLet Y = X1 + X2 + … + Xn•Y is distributed chi-squared with degrees of freedom Denoted as  2(v)nvvvv  214Linear Combinations of R.V.’sX1, X2, … , Xn are identically distributed normal r.v.’s with mean  and variance  2Let •Y is distributed chi-squared with n degrees of freedom, denoted as  2(n)niiniiZXY12125SamplingParameter•A property of an probability distribution.•In real life, these are typically unknown.Population•The totality of the observations of interest.•Typically, it is not practical to see all observations in a population.Sample•A subset of observations from a population.•These should be representative of the population.6SamplingRandom Sample•Before data: X1, X2, … , Xn are independent and identically distributed (i.i.d.) r.v.’s•After data: Denote observed values by x1, x2, …, xn7EstimatorsA statistic is a function u(X1, X2, … , Xn)An estimator is a statistic designed to estimate an unknown parameter• is an estimator for the population mean • is an estimator for the population median•S2 is an estimator for the population variance  2Statistics and estimators are r.v.’sAn estimate is a calculated value (not random) of an estimator•e.g., , , s2XX~~xx~8Sampling DistributionsBefore we collect data, the value of a statistic u(X1, X2, … , Xn) is randomIf we collect multiple random samples: •Sample 1  •Sample 2  •etc.This variability is called sampling errorA sampling distribution is the distribution for a statistic2.3x9.2x9Sampling Distributions - SetupLet X denote an observation to be sampled•X follows the population distributionIf we collect a sample of n observations•X1, X2, … , Xn are i.i.d. population distribution•E[Xi] =  = the population mean•V(Xi) =  2 = the population varianceExample: Suppose we are studying the lifetime of lightbulbs, and we collect n = 100•Assume the population distribution is Exp()•X1, X2, … , X100 are i.i.d. Exp()10Sampling Distributions - MeanSample Mean (before data)Expected value of         )(1111111112121nnnXEnXEnXEnXEnXXXnEXEniniinn nniiXXXnXnX 21111X11Sampling Distributions - MeanVariance of•Note: X1, X2, … , Xn are independent r.v.’s        nnnnXVnXVnXVnXVnXXXnVXVniniinn222122122221221)(1111111X12Sampling Distributions - MeanIf the population distribution is N( ,  2)•X1, X2, … , Xn are i.i.d. N( ,  2)• is distributed N( ,  2/n)• is distributed N(0 , 1)Example: X1, …, X25 i.i.d. N( = 15 ,  2 = 100)XnXZ/ 9938.0)5.2(5/101520/20 nXPXP13Central Limit Theorem (CLT)X1, …, Xn i.i.d some population distribution with mean  and variance  2For large n• is approximately distributed N( ,  2/n)Example: X1, …, X64 i.i.d. Exp( = 4) = E[Xi] =  = 4 ,  2 = V(Xi) =  2 = 16X  0228.0)2(128/443/3ZPnXPXP14Sampling Distributions - MeansTwo Independent Populations (any distributions)•Pop. 1  Sample 1 with n1 obsns  •Pop. 2  Sample 2 with n2 obsns  Expected value of Variance of •Note: and are independent r.v.’s1X2X     212121 XEXEXXE21XX 21XX      2221212121nnXVXVXXV1X2X),(211),(22215Sampling Distributions - MeansIf Pop. 1 is and Pop. 2 is • is distributedCLT: For large n1 and n2 and for any distns, if Pop. 1 has mean 1 and variance , and Pop. 2 has mean 2 and variance • is approximately distributed),(211N),(222N21XX 22212121,nnN212221XX 22212121,nnN16Sampling Distributions - MeansExample: Pop. 1 is distributed Exp( = 4) and Pop. 2 is distributed Gamma( = 3,  = 4) 1 =  = 4 , 2 =  = 12 , Sample 64 obsns from Pop.1 and 75 obsns from Pop.2 1  2 = 8• 89.075486416222121nn162214822217Sampling Distributions - Means  9830.0)12.2(12.289.0)8(6)(6222121212121ZPnnXXPXXP18Sampling Distributions - VarianceSample Variance (before data)•Note:Assume X1, X2, … , Xn are i.i.d. N( ,  2)•Then is distributed  2(n  1) 22)1(Sn niiXXnS122)(11 212)1( SnXXnii19Sampling Distributions - VarianceRecall is distributed  2(n)Now