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chap05PRN econ 325

With microsoft excel normal distribution

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With Microsoft Excel normal distribution probabilities can be obtained by selecting Insert Function NORMDIST. The general usage is: NORMDIST(x, X X , σ μ , cumulative) where cumulative = 0 for the PDF, cumulative = 1 for the CDF
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Econ 325 – Chapter 5 15 Working with the Normal Distribution Before the days of high speed laptop computers, applied workers used statistical tables (printed in the Appendix to statistical textbooks) to look-up normal distribution probabilities. Working with the statistical tables can be useful as a learning exercise as it gives emphasis to understanding the properties of the normal distribution. Therefore, as a check on the calculations that can be obtained with Microsoft Excel, the use of the normal distribution tables will be described here. It can be noted that probabilities depend on the setting of X μ and X σ , the mean and standard deviation of the random variable. However, it turns out that probabilities for the standard normal random variable Z with mean 0 and variance 1 can be used to calculate probabilities for any other normal distribution. Textbooks provide an appendix table for the cumulative distribution function (CDF) for the standard normal random variable: ) 1 , 0 ( N ~ X Z X X σ μ - = Econ 325 – Chapter 5 16 How is the table read ? A graph is useful. Probability Density Function (PDF) of Z z0 0 f(z) z Area = P(Z < z0) = F(z0) Area = 1 - F(z0) For a value of interest 0 z the table gives the cumulative probability: ) z Z ( P ) z ( F 0 0 = The table lists values for 0 z 0 only. From symmetry of the normal distribution: ) z ( F 1 ) z Z ( P ) z Z ( P ) z ( F 0 0 0 0 - = = - = -
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Econ 325 – Chapter 5 17 A result for a range probability with symmetric upper and lower values can be stated. For some value 0 z : 1 ) z ( F 2 )] z ( F 1 [ ) z ( F ) z Z ( P ) z Z ( P ) z Z z ( P 0 0 0 0 0 0 0 - = - - = - - = - This is shown with a graph. PDF of Z z0 0 -z0 f(z) z Area = F(z0) - F(-z0) Upper Tail Area = 1 - F(z0) Lower Tail Area = F(-z0) By symmetry of the normal distribution the area in the “lower tail” is identical to the area in the “upper tail.” Econ 325 – Chapter 5 18 Now suppose the random variable to work with is: ) , ( N ~ X 2 σ μ For two numerical values a and b , with a < b , a probability of interest is: ) b X a ( P < < This probability statement can be transformed to a probability statement about the standard normal random variable Z . This is done as follows: σ μ - - σ μ - = σ μ - < < σ μ - = σ μ - < σ μ - < σ μ - = < < a F b F b Z a P b X a P ) b X a ( P The Appendix Table can now be used to look-up the cumulative probabilities for the standard normal distribution.
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Econ 325 – Chapter 5 19 Example Let the continuous random variable X be the amount of money spent on clothing in a year by a university student. It is known that: ) , ( N ~ X 2 σ μ with $380 = μ and 50 $ = σ Questions and Answers square4 Find ) X ( P 400 < . This gives the probability that a randomly chosen student will spend less than $400 on clothing in a year. First state the probability as a probability about the standard normal variable Z : ) ( F Z ( P Z P X P ) X ( P 0.4 0.4) 50 380 400 400 400 = < = - < = σ μ - < σ μ - = < Now look-up the answer in the Appendix Table.
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