chap07PRN econ 325

In general find random variables low ˆ θ and high

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In general, find random variables low ˆ θ and high ˆ θ such that: α - = θ < θ < θ 1 ) ˆ ˆ ( P high low Greek letter alpha α - 1 is called the confidence level . ] ˆ , ˆ [ high low θ θ gives a ) 1 ( α - 100 % confidence interval estimator for θ . Econ 325 – Chapter 7 10 x Interval Estimation for the Population Mean Results established in previous lecture notes are first reviewed. A point estimator for the population mean μ is: = = n 1 i i X n 1 X X has a sampling distribution with the properties: μ = ) X ( E ( X is an unbiased estimator of μ ) and n ) X ( Var 2 σ = The standard error of X is: n ) X ( se σ = To proceed further, assume that X follows a normal distribution. This assumption is reasonable since: If 1 X , 2 X , . . . , n X follow a normal distribution then a result is that X also has a normal distribution. Even if 1 X , 2 X , . . . , n X are not normally distributed, by the Central Limit Theorem, X will tend to the normal distribution. A standard normal random variable is: ) 1 , 0 ( N ~ n X ) X ( se X Z σ μ - = μ - =
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Econ 325 – Chapter 7 11 A procedure for interval estimation is now developed. A critical value c z can be found so that: α - = < < - 1 ) z Z z ( P c c where α - 1 is set to a desired level. Rearrange the probability statement to obtain: ( ) n z X n z X P z n X z P ) z Z z ( P c c c c c c σ + < μ < σ - = < σ μ - < - = < < - This gives the ) 1 ( α - 100 % confidence interval estimator for the population mean μ as: [ n z X , n z X c c σ + σ - ] This can be written as: n z X c σ ± Econ 325 – Chapter 7 12 To finish off, the critical value c z must be set. For a 90% confidence interval estimator, with 0.10 = α , find a number c z such that: 90 . 0 ) z Z z ( P c c = < < - An illustration is below. PDF of Z zc 0 -zc f(z) z Area = 0.9 Upper Tail Area = 0.05 Lower Tail Area = 0.05 Note that the area in each tail is: 0.05 2 0.10 = = α 2
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Econ 325 – Chapter 7 13 This says find c z such that: 95 . 0 2 1 ) z ( F ) z Z ( P c c = α - = = < The Appendix Table for the standard normal distribution can be used to get an answer. Another option is to use Microsoft Excel. Select Insert Function: NORMSINV( 0.95 ) This returns the answer: 1.645 = c z The confidence level α - 1 can be set to any desired probability. A table of some popular choices is below. Confidence level 2 α Microsoft Excel NORMSINV probability c z 0.90 0.05 0.95 1.645 0.95 0.025 0.975 1.96 0.99 0.005 0.995 2.576 Econ 325 – Chapter 7 14 An application can now proceed. Collect a data set with numeric observations: 1 x , 2 x , . . . , n x . The point estimate for the population mean μ is the calculated sample mean: = = n 1 i i x n 1 x Assume that the population standard deviation σ is known from previous research.
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In general find random variables low ˆ θ and high ˆ θ...

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