Modeling Continuous VariablesModelsExamplesSlide 4Example of a ModelHistograms and AreaSlide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Density FunctionsSlide 15Slide 16Slide 17Slide 18Slide 19The Normal DistributionSlide 21Slide 22Slide 23Slide 24Slide 25Slide 26Some Normal DistributionsSlide 28Slide 29Slide 30Bag A vs. Bag BSlide 32Slide 33Slide 34Slide 35Slide 36Slide 37Slide 38Slide 39Slide 40Slide 41Slide 42Slide 43Slide 44Slide 45Slide 46Modeling Modeling Continuous Continuous VariablesVariablesLecture 19Lecture 19Section 6.1 - 6.3.1Section 6.1 - 6.3.1Fri, Feb 24, 2006Fri, Feb 24, 2006ModelsModelsMathematical modelMathematical model – An – An abstraction and, therefore, a abstraction and, therefore, a simplification of a real situation, one simplification of a real situation, one that retains the essential features.that retains the essential features.Real situations are usually much to Real situations are usually much to complicated to deal with in all their complicated to deal with in all their details.details.ExamplesExamplesEconomic models treat money as a Economic models treat money as a continuous quantity, even though it is continuous quantity, even though it is discrete.discrete.This is an abstraction that is incorporated into This is an abstraction that is incorporated into the model to make it simpler.the model to make it simpler.The “bell curve” is a model (an abstraction) The “bell curve” is a model (an abstraction) of many populations.of many populations.Real populations have all sorts of bumps and Real populations have all sorts of bumps and twists.twists.The bell curve is smooth and perfectly The bell curve is smooth and perfectly symmetric.symmetric.ModelsModelsNo mathematical model is perfect.No mathematical model is perfect.A mathematical model is useful (and A mathematical model is useful (and powerful) to the extent that it is a powerful) to the extent that it is a faithful representation of reality.faithful representation of reality.Conversely, to the extent that is it Conversely, to the extent that is it not faithful to reality, it can lead to not faithful to reality, it can lead to false conclusions about the situation false conclusions about the situation that it is supposed to model.that it is supposed to model.Example of a ModelExample of a ModelUse a random number generator to Use a random number generator to simulate how a pair of rolled dice will land.simulate how a pair of rolled dice will land.The possible totals range from 2 to 12.The possible totals range from 2 to 12.Using the TI-83, which is a correct model?Using the TI-83, which is a correct model?Enter Enter randInt(2, 12)randInt(2, 12), that is, get a random , that is, get a random number from 2 to 12, ornumber from 2 to 12, orEnter Enter 2*randInt(1, 6)2*randInt(1, 6), that is, double a random , that is, double a random number from 1 to 6, ornumber from 1 to 6, orEnter Enter randInt(1, 6) + randInt(1, 6)randInt(1, 6) + randInt(1, 6), that is, add , that is, add two random numbers from 1 to 6.two random numbers from 1 to 6.Histograms and AreaHistograms and AreaIf a histogram is drawn If a histogram is drawn appropriately, then frequency is appropriately, then frequency is represented by area.represented by area.Consider the following histogram of Consider the following histogram of test scores.test scores.GradeGradeFrequencFrequencyy60 – 6960 – 693370 – 7970 – 798880 – 8980 – 899990 – 9990 – 9955Histograms and AreaHistograms and AreaGradeFrequency607080 901000246810Histograms and AreaHistograms and AreaIn the histogram, we may replace In the histogram, we may replace the frequency with the proportion the frequency with the proportion (of the total).(of the total).GradeGradeFrequencyFrequencyProportionProportion60 – 6960 – 69330.120.1270 – 7970 – 79880.320.3280 – 8980 – 89990.360.3690 – 9990 – 99550.200.20Histograms and AreaHistograms and AreaGradeProportion607080 9010000.100.200.300.40Histograms and AreaHistograms and AreaGradeProportion607080 9010000.100.200.300.40Histograms and AreaHistograms and AreaFurthermore, we may divide the Furthermore, we may divide the proportions by the width of the proportions by the width of the classes to get the classes to get the densitydensity..GradeGradeFrequencyFrequencyProportionProportionDensityDensity60 – 6960 – 69330.120.120.0120.01270 – 7970 – 79880.320.320.0320.03280 – 8980 – 89990.360.360.0360.03690 – 9990 – 99550.200.200.0200.020Histograms and AreaHistograms and AreaGradeDensity607080 9010000.0100.0200.0300.040Histograms and AreaHistograms and AreaThe final histogram has the special The final histogram has the special property that the property that the proportionproportion can be can be found by computing the found by computing the areaarea of the of the rectangle.rectangle.The vertical scale has been adjusted so The vertical scale has been adjusted so that the total area is 1, or 100%.that the total area is 1, or 100%.For example, what proportion of the For example, what proportion of the grades are less than 80?grades are less than 80?Compute: (10 Compute: (10 0.012) + (10 0.012) + (10 0.032) 0.032)= 0.12 + 0.32 = 0.44 = 44%.= 0.12 + 0.32 = 0.44 = 44%.Density FunctionsDensity FunctionsThis is the fundamental property This is the fundamental property that connects the graph of a that connects the graph of a continuous model to the population continuous model to the population that it represents, namely:that it represents, namely:The The area under the grapharea under the graph between two between two numbers numbers aa and and bb on the on the xx-axis -axis represents the represents the proportion of the proportion of the populationpopulation that lies between that lies between aa and and bb.. AREA = PROPORTIONAREA = PROPORTIONDensity FunctionsDensity FunctionsThe area under the curve between The area under the curve between aa and and bb is the proportion of the values is the proportion of the values of of xx that lie between that lie between aa and and bb..a bxDensity FunctionsDensity FunctionsThe area under the curve between The area under the curve between aa and and bb is the proportion of the values is the proportion of the values of of xx that lie between that lie between aa and and bb..a bxDensity FunctionsDensity FunctionsThe area under the curve
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