chap07PRN econ 325

Even if 1 x 2 x n x are not normally distributed by

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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. A 90% confidence interval estimate is calculated as: n x σ ± 1.645 head2right What is the interpretation of an interval estimate ? In applied work, the calculated interval estimate is based on one sample of data. It may contain the true parameter μ , or it may not contain μ . Since the true value of μ is unknown, it is impossible to say whether or not the population mean is contained in the interval estimate calculated from the sample of data.
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Econ 325 – Chapter 7 15 The interpretation of a 90 % confidence interval estimate for the population mean can be explained in the context of repeated sampling. Different samples of data will give different calculated sample means. Some of the sample means will be less than the true mean μ and some will be greater than this value. Therefore, different samples will give different interval estimates. In a ‘large’ number of samples 90% of the interval estimates will contain the true population mean μ and the other 10% will not contain this value (that is, they either have an upper limit below μ or a lower limit that exceeds μ ). Econ 325 – Chapter 7 16 The interpretation of a 90 % confidence interval estimate can be demonstrated with a computer simulation.
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