ECE 307 – Techniques for Engineering Decisions 13. Data UsesFOCUS OF DATA USAGE TOPICEXAMPLEUSE OF HISTOGRAMSCONSTRUCTION OF THE c.d.f.STATISTICAL PARAMETER ESTIMATORSSTATISTICAL PARAMETER ESTIMATORSSTATISTICAL PARAMETER ESTIMATORS STATISTICAL PARAMETER ESTIMATORS STATISTICAL PARAMETER ESTIMATORSSTATISTICAL PARAMETER ESTIMATORSEXAMPLE: TACO SHELLSEXAMPLE: TACO SHELLSEXAMPLE: TACO SHELLSUNBROKEN TACO SHELLSEXAMPLE: TACO SHELLSc.d.f.s CONSTRUCTED FOR THE TWO SUPPLIERS c.d.f.s OF THE TWO SUPPLIERSc.d.f.s OF THE TWO SUPPLIERSREDUCED ORDER REPRESENTATION OF THE TEST RUN DATACOMMENTSCOMMENTS© 2006 – 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 1ECE 307 – Techniques for Engineering Decisions13. Data UsesGeorge GrossDepartment of Electrical and Computer EngineeringUniversity of Illinois at Urbana–Champaign© 2006 – 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 2 Use of historical data for the construction of probability distributions The interpretation of probability information Use of estimators Application exampleFOCUS OF DATA USAGE TOPIC© 2006 – 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 3EXAMPLE Consider the interpretation of the statement We obtain this probability from, say, 20 years of June weather data in Champaign, with each day classified as either sunny or not sunny The 600 June days of data indicate that 318 or 53 %of these days are classified as sunny Given the long– term historical behavior in the data, the probability of 0.53 makes sense© 2006 – 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 4outage capacity of a generation plant (MW )USE OF HISTOGRAMSrated capacity 0 outagefull outage capacityrelative frequency (days/3650)highderated capacitylowderated capacity© 2006 – 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 5CONSTRUCTION OF THE c.d.f.1.0apx© 2006 – 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 6STATISTICAL PARAMETER ESTIMATORS An estimator is a r.v. that can be used to estimate the value of a parameter of interest Consider a r.v. whose statistical parameters we wish to estimate We consider a set of r.v.s , where each is independent of , , andeach has the same distribution as ; we refer to this set as a set of n independent, identically distributed or i.i.d. r.v.s© 2006 – 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 7STATISTICAL PARAMETER ESTIMATORS We use the set of n i.i.d. r.v.s to construct estimators for the moments of  We focus on the estimators for two key parameters of :  the mean of  the variance of© 2006 – 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 8 The sample mean estimator is the r.v. In practice, we obtain an estimate of the mean by using the observed realizations of the n r.v.sSTATISTICAL PARAMETER ESTIMATORS© 2006 – 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 9 The estimator of the sample variance is given by the r.v. We obtain an estimate of the variance by using the observed realizations of the n r.v.sSTATISTICAL PARAMETER ESTIMATORS© 2006 – 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 10STATISTICAL PARAMETER ESTIMATORS An equivalent way to think about the computation of the estimate is to draw n random samples from the sample space of  We collect the set of n random samples of the r.v. : these are n randomlydrawn values from the sample space of© 2006 – 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 11STATISTICAL PARAMETER ESTIMATORS The value computed with the set of random samples provides an estimate of  The value s2computed with the set of random samples provides an estimate of© 2006 – 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 12EXAMPLE: TACO SHELLS This application example focuses on taco shells and is concerned with the high breakage rate in the shipment of most taco shells: typical rate is 10 – 15 % A company with a new shipping container claims to have a lower – approximately 5% – breakage rate This company’s price is $ 25 for a 500–taco shell box vs. $ 23.75 for a 500–taco shell box of the current supplier© 2006 – 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 13EXAMPLE: TACO SHELLS A test run using 12 boxes from the new company and 18 boxes from the current company is performed and used for comparison purposes: we randomly pick the elements to construct the set from the sample space of the r.v. to represent© 2006 – 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 14EXAMPLE: TACO SHELLSthe number of unbroken shells from the new company and the elements to construct the set from the sample space of the r.v. to represent those of the current company We tabulate the data of the useable shells from the two suppliers© 2006 – 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 15UNBROKEN TACO SHELLSnew supplier current supplier468 467 444 441 450474 469 449 434 444474 484 443 427 433479 470 440 446 441482 463 439 452 436478 468 448 442 429© 2006 – 2021 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved. 16EXAMPLE: TACO SHELLS12 samples of thenumber of unbroken shells (x)18 samples of thenumber of unbroken shells (y)costs per