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# Class Note for EMSE 273 with Professor Dorp at GW (8)

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Date Created: 02/07/15
EMSE 273 Discrete System Simulation An Explanation of Confidence Intervals Half Widths and Credibility Intervals Let X 1 X be an independent random sample from the random variable X De ne the sample average of such a random sample as and the sample variance as n1i1 Note that both X and A2 are random variables themselves Indeed every time we draw a new sample realization 1 ZUn these values Changes as well Lecture Notes by Dr JR van Dorp Page 1 EMSE 273 Discrete System Simulation Let J be the true mean of the random variable X and let us consider the statistic T X M T X Xn 1 A n This is called a Statistic since it is a function of the random sample X1 X We shall write A 2 02 but please be mindful of the fact that i1 the sample standard deviation A is also a function of the random sample X1Xn The sample average sample variance and sample standard deviation are also statistics of the sample X1 Xn Each one provides specific information about this sample Lecture Notes by Dr JR van Dorp Page 2 EMSE 273 Discrete System Simulation It can be shown that the distribution of TX1 X follows a student t distribution with n 1 degrees of freedom if the X1 Xn are drawn from a normal distribution Slight deviations from normality are allowable but if the X1 X n are drawn from a highly non normal distribution the T statistic should not be used if the sample size is small which is usually not the case in simulation Similar to the standard normal distribution the student t distribution is symmetric around zero In fact for large sample sizes the student t distribution converges to the standard normal distribution De ne tn171a2 as follows a PT TX1Xn S tn171a2 1 5 Hence if we set 04 5 we have PT TX1 Xn S tn170975 97500 ltgt PT TX1 Xn gt tn170975 2500 Lecture Notes by Dr JR van Dorp Page 3 EMSE 273 Discrete System Simulation tn IJ aZ tn 11 02 0 PDF of TX1Xn 05 05 PrTX1939 9Xn S tn 11 a2 3 PrTX1939 9Xn gttn 11 a2 3 MicroSoft EXCEL has a function such that TINVan 1 tn171a2 Lecture Notes by Dr JR van Dorp Page 4 EMSE 273 Discrete System Simulation Because of symmetry of the student If distribution we also have PT TX1 Xn S tn170975 2500 ltgt PT TX1 Xn gt tn170975 97500 Therefore PT tn171a2 S TltX1 Xn Stn171a2 1 06 Recalling the definition X M TX17 7X71 An we have X PM tn 11 oz2 S A M S tn l 1 a2 1 04 ltgt Lecture Notes by Dr JR van Dorp Page 5 EMSE 273 Discrete System Simulation tn 11 a2 X A n Pr The interval X tn 11 a2 X A X tn 11 a2 X A quot quot is called a 1 a con dence interval for the true value J of mean of the random variable X Lecture Notes by Dr JR van Dorp Page 6 EMSE 273 Discrete System Simulation The halfwidth of the 1 a con dence interval is defined as tn 11 a2 X A21 7 n Since the value J is a constant we know however that 1 ME ab PTCU E 05 0 u Gab for any fixed a and I such that a lt 9 So why do we call the interval X tn 11 a2 X A X tn 11 a2 X A x x a 1 00 confidence interval for the value of M Lecture Notes by Dr JR van Dorp Page 7 EMSE 273 Discrete System Simulation Answer The lower bound L1a X tn 11 a2 X A n and the upper bound tn 11 a2 X A n are in fact random variables themselves and not fixed and PrJ E L1a U1a 1 a Ul a Note that in the above probability statement L1a and U1a are the random variables not H Hence a 1 oz con dence interval is a randomly changing interval that 1 oz of the times captures the mean value 4 Lecture Notes by Dr JR van Dorp Page 8 EMSE 273 Discrete System Simulation Let x1 zcn now be a realization of the random sample X1 Xn Hence we can now calculate a realization E of the random sample mean X and a realization 039 of the random sample variance A We can also now calculate a realization l of the random lower bound Lla and a realization u of the random upper bound U1a The interval l u is now almost always presented as a 1 00 con dence interval of the mean value M However since l and u are fixed the only meaningful conclusion we can draw is PM E MD 1 Thus l u should be presented as a realization of the random 1 00 confidence interval L1a U1a Unfortunately this subtle difference but very important difference often gets lost in the presentation of analysis results Even with the proper presentation a lay person not trained in statistics will likely interpret the interval U u such that P39r39J E l 1 a which we know not to be the case Lecture Notes by Dr JR van Dorp Page 9 EMSE 273 Discrete System Simulation Credibility Intervals Given an estimated empirical cumulative distribution XCU constructed from an sample 1 it we can establish an p X 100 credibility interval 1 U such that PrX e LU p by setting L lt1 pgt2 13 10 U 1 lt1 pgt2 13 10 1 where U1p2 and x11p2 are quantiles of the empirical distribution function For example ifn 100 and if we set 10 090 we have 005 5 The 5 th order statistic 095 035 The 95 th order statistic Lecture Notes by Dr JR van Dorp Page 10 EMSE 273 Discrete System Simulation Review Confidence Intervals Confidence intervals are calculated for characteristics of X such as for example EX Va7 X etc For a random sample X1 Xn the probability that an 1 oz100 confidence interval which is a random interval for E X captures EX equals 1 04 For a fixed sample 1 it no probability interpretation can be assigned to a realization of an 1 04100 confidence interval When the sample size n increases the width of confidence intervals decrease They converge to a true value a single point Lecture Notes by Dr J R van Dorp Page 11 EMSE 273 Discrete System Simulation Review Credibility Intervals Credibility intervals are calculated for the random variable X For a random sample X1 Xn the probability that X is a member within an 1 04100 credibility interval for X which is also random interval equals 1 04 When the sample size n increases the width of credibility intervals do not decrease Credibility intervals converge to the true probability interval For a fixed sample 1 it the probability that X is a member within an 1 oz100 credibility interval for X which is also random interval equals approximately 1 oz Lecture Notes by Dr JR van Dorp Page 12 EMSE 273 Discrete System Simulation HOMEWORK 1 1 Develop an Excel Macro called quotSaveWaitInQueueProbabilityquot that saves 365 replications of the quotWait in Queue Probabilityquot in the work sheet quotWait in Queue Probabilityquot in the Spreadsheet quotSimpleQueueCon dencexlsquot 2 Plot a 20 point approximation of the cumulative distribution function of quotWait in Queue Probabilityquot in the worksheet quotWait in Queue Probabilityquot 3 Plot a 20 point histogram of the probability density function of quotWait in Queue Probabilityquot in the worksheet quotWait in Queue Probabilityquot 4 Fill in the template cells in the worksheet quotWait in Queue Probabilityquot to calculate 95 credibility and 95 confidence intervals for the quotWait in Queue Probabilityquot Lecture Notes by Dr JR van Dorp Page 13 EMSE 273 Discrete System Simulation HOMEWORK 2 1 Develop an Excel Macro called quotSaveMaxInSystemquot that saves 365 replications of the quotMax Customers In The Systemquot in the work sheet quotMaximum Customers in the Systemquot in the Spreadsheet quotSimpleQueueCon dencexlsquot 2 Plot a 10 point approximation of the cumulative distribution function of quotMax Customers In The Systemquot in the worksheet quotMaximum Customers in Systemquot 3 Plot a 10 point histogram of the probability mass function of quotMax Customers In The Systemquot in the worksheet quotMaximum Customers in Systemquot 4 Fill in the template cells in the worksheet quotMaximum Customers in Systemquot to calculate 95 credibility and 95 confidence intervals for the quotMax Customers In The Systemquot Lecture Notes by Dr JR van Dorp Page 14

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