The sampling distribution of the sample mean l are

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the sampling distribution of the sample mean l Are there circumstances where the exact sampling distribution is known? l Answer is yes in certain cases l Consider 2 cases l Population distribution of X is normal l Population distribution of X is non-normal & n is large
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16 Sampling from a normal population l If X~N ( μ , σ 2) then X i ~N ( μ , σ 2) l Sample mean is also normal as it is a linear combination of normal rv’s l Let X represent outstanding balances of customers of a firm l From past experience X is well approximated by a normal distribution with μ = $40 & σ = $10 l If an auditor takes a random sample of 25 accounts what is the probability that the mean balance would be less than $36? 0228 . ) 2 0 ( 5 . ) 2 ( ) 2 ( 2 40 36 ) 36 ( ) 4 , 40 ( ~ ) 100 , 40 ( ~ = < < - = = - < = - < - = < Z P Z P Z P n X P X P N X N X σ μ
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17 Sampling from a non-normal population l For teller example population distribution of X is clearly non-normal l X is discrete taking on only 3 values of 0, 1, 2 l X is skewed to the right l With sample size of 2 the sampling distribution of the sample mean remains non- normal l What happens as you increase the sample size? x 0 1 2 P ( X = x ) 3/5 1/5 1/5 0 1/2 1 3/2 2 9/25 6/25 7/25 2/25 1/25
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Central Limit Theorem (CLT) l The sampling distribution of the mean of a random sample drawn from any population with mean μ & variance σ 2 will be approximately normally distributed for a sufficiently large sample size l Based upon a limiting argument we have that 18 ) / , ( ~ big ly sufficient for ely, approximat or, ) 1 , 0 ( / ) ( 2 n N X n N X n a L σ μ σ μ → -
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Central Limit Theorem... l How large is “sufficiently large” is an empirical matter l Will depend on non-normality of the population distribution l From a modelling perspective a sufficiently large sample implies it is not necessary to assume normality of the underlying population to make inferences about the sample mean based upon the normal distribution l This is an extremely powerful & useful implication 19
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20 Auditing accounts l Clare is an auditor l A client has customer accounts with outstanding balances that are known to have μ = $30.3, σ = $30.334 l Clare decides on a random sample of 250 accounts l What is the probability that the sample mean balance will be within $4 of the population mean balance? 9624 . ) 08 . 2 08 . 2 ( 92 . 1 3 . 30 3 . 34 / 92 . 1 3 . 30 3 . 26 ) 3 . 34 3 . 26 ( ) 92 . 1 , 3 . 30 ( ~ or ) / , ( ~ CLT By CLT) the in ' ' the ignore ll we' so (No, matter? it Does normal? Is of on distributi sampling requires which ) 3 . 34 3 . 26 ( determine to Need 2 2 a = < < - = - < - < - = < < < < Z P n X P X P N X n N X X X X P σ μ σ μ
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Auditing accounts… l According to our results Clare will almost always (96% chance) obtain a sample mean within $4 of the population mean (designated task) l Problem could have been solved without knowledge of μ l Auditor has a measure of sampling error for the sample mean as an estimate for an unknown population mean 21 9624 . ) 08 . 2 08 . 2 ( 92 . 1 4 / 92 . 1 4 ) 4 4 ( )) 92 . 1 , ( ~ (or ) / , ( ~ CLT By large size sample because matter t Doesn' normal? Is of on distributi sampling requires which ) 4 4 ( determine to Need 2 2 = < < - = < - < - = + < < - + < < - Z P n X P X P N X n N X X X X P σ μ μ μ μ σ μ μ μ
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Auditing accounts...
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