Stat_Chap8

Stat_Chap8 - Confidence Intervals Chapter 8, all sections...

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Confidence Intervals Chapter 8, all sections except 8-7 1

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Estimate population mean Estimate population mean using sample mean Accuracy/precision: sampling distribution (CLT or normal population) Measure of precision: standard error 2 2 ~( , / ) X Nn μσ : / SE n σ
Constructing an interval Recall Construct an interval that contains the true mean with given probability || 3 ( 3 ) ( 3 ) 9 9 . 7 % PX n σ μ ⎛⎞ −≤ = Φ Φ = ⎜⎟ ⎝⎠ 99.7% 3 , 3 = + n X n X P σσ 3

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Look for an interval that contains m with probability 1- α z a /2 : percentage point of N(0,1) α σ μ = 1 / 2 / 2 / z n X z P ? X n ± 4 || ? ( ? ) ( ? ) 1 - PX n ⎛⎞ −≤ = Φ Φ −= ⎜⎟ ⎝⎠
5 0 - z α /2 z α /2 N(0,1) α /2 α /2 1- α /2 0.05 0.05 0.025 0.16 () / 2 0.1: ( ) 5%, 1.645 qnorm(0.05) 1.645 0.05: 1.96; 0.32 : 0.99 z zz z α αα Φ− = = = =− == = =

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The following interval contains the true mean with probability 1- α α = 10%, the following interval contains the true mean with probability 90% is a random variable . After obtaining a numerical value , obtain a numerical interval ] / , / [ 2 / 2 / n z X n z X σ α + ] / 645 . 1 , / 645 . 1 [ n X n X + X x 6
Confidence interval Confidence interval (CI) : given the sample mean of a random sample, a 100(1- α ) % CI on the population mean μ is given by 100(1- α ) %: confidence level ] / , / [ 2 / 2 / n z x n z x σ α + x 7

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