# L given a population distribution of errors the

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Errors by bank tellers... l Given a population distribution of errors the population parameters could be determined l This is unknown to the supervisor, of course! l Let the population distribution be given by: 9 x 0 1 2 P ( X = x ) 3/5 1/5 1/5 25 16 5 1 5 3 2 5 1 5 3 1 5 3 5 3 0 ) ( 5 3 5 1 2 5 1 1 5 3 0 ) ( Then 2 2 2 2 = × - + × - + × - = = = × + × + × = = X Var X E σ μ

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10 Errors by bank tellers… l What is the sampling distribution of the sample mean for n =2? l Need all possible samples of size 2 l For each X 1, X 2 pair need to calculate sample mean & associated probability l Now have the distribution for this new random variable Sample X Probability 0, 0 0 9/25 0 ,1 1/2 3/25 0, 2 1 3/25 1, 0 1/2 3/25 1, 1 1 1/25 1, 2 3/2 1/25 2, 0 1 3/25 2, 1 3/2 1/25 2, 2 2 1/25 x 0 1/2 1 3/2 2 ) ( x X P = 9/25 6/25 7/25 2/25 1/25
11 Errors by bank tellers… l Note: l Sampling distribution of the sample mean could be obtained in similar manner for other sample sizes l Could also produce sampling distribution for sample variance (or any other statistic) l Have derived sampling distribution directly - in general this isn’t necessary Sample s 2 Probability 0, 0 0 9/25 0 ,1 1/2 3/25 0, 2 2 3/25 1, 0 1/2 3/25 1, 1 0 1/25 1, 2 1/2 1/25 2, 0 2 3/25 2, 1 1/2 1/25 2, 2 0 1/25 s 2 0 1/2 2 P ( S 2 =s 2 ) 11/25 8/25 6/25

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12 Errors by bank tellers… l Can now use the sampling distribution to derive expectations & calculate probabilities l What are the mean & variance of the sampling distribution of the sample mean? l How do they relate to the population mean & variance? l Recall that μ = E ( X )=3/5 & σ 2= Var( X )=16/25 = = - + + - = = = + + + + = n X Var X E X X 2 2 2 2 25 8 25 1 5 3 2 25 9 5 3 0 ) ( ) ( 5 3 25 1 2 25 2 2 3 25 7 1 25 6 2 1 25 9 0 ) ( σ σ μ μ
13 Properties of the sampling distribution of the sample mean l Standard deviation of the sampling distribution is called the standard error (se) l Relationships found in teller example are general ones l What do these properties imply about the sample mean as an estimator of the population mean? n X Var X se n n X Var X Var n X E X E i i / ) ( ) ( ) ( ) ( prove Can 2 σ σ μ = = = = = =

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14 Different means – some clarification l Let X be rv of interest l It has a population mean l Sample mean is a rv l It is a function of a sample of size n l Sample mean has a sampling distribution l Which also has a (population) mean l Mean of the sampling distribution of the sample mean is equal to the population mean l A particular sample will yield a specific value for the sample mean x X X E X n X X E(X) μ i = = = = : sample particular a In ) ( : of on distributi sampling of Mean : mean Sample : mean Population μ
15 Sampling distribution of the sample mean l Have general expression for mean & variance of

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