Homework ReviewModelsArea And ProbabilityDensity FunctionsThe Normal DistributionsExamplesAssignmentModelingContinuousVariablesRobb T.KoetherHomeworkReviewModelsArea AndProbabilityDensity FunctionsThe NormalDistributionsExamplesAssignmentModeling Continuous VariablesLecture 19Sections 6.1 - 6.3.1Robb T. KoetherHampden-Sydney CollegeTue, Feb 17, 2009ModelingContinuousVariablesRobb T.KoetherHomeworkReviewModelsArea AndProbabilityDensity FunctionsThe NormalDistributionsExamplesAssignmentOutline1Homework Review2Models3Area And ProbabilityDensity Functions4The Normal DistributionsExamples5AssignmentModelingContinuousVariablesRobb T.KoetherHomeworkReviewModelsArea AndProbabilityDensity FunctionsThe NormalDistributionsExamplesAssignmentHomework ReviewExercise 5.20, page 336.Consider the following two frequency plots based on twosets of data with 10 observations:-2 210-1Frequency Plot 1-2 210-1Frequency Plot 2Note that Plot 1 can be converted to Plot 2 by moving thetwo blue X’s.ModelingContinuousVariablesRobb T.KoetherHomeworkReviewModelsArea AndProbabilityDensity FunctionsThe NormalDistributionsExamplesAssignmentHomework ReviewExercise 5.20, page 336.(a) Do you think the range for Plot 1 is smaller than, equalto, or larger than the range for Plot 2?(b) Do you think the mean for Plot 1 is smaller than, equalto, or larger than the mean for Plot 2?(c) Do you think the standard deviation for Plot 1 is smallerthan, equal to, or larger than the standard deviation forPlot 2? Explain.(d) Check your answers to parts (a), (b), and (c) bycomputing the range, mean, and standard deviation foreach set of data.ModelingContinuousVariablesRobb T.KoetherHomeworkReviewModelsArea AndProbabilityDensity FunctionsThe NormalDistributionsExamplesAssignmentHomework ReviewSolution(a) The ranges are clearly equal.(b) Both distributions are symmetric, so the means are inthe center, at 0. Thus, they are equal.(c) The two blue points in Plot 2 are closer to the mean thanthey are in Plot 1, so they have smaller deviations. Allother points are the same between the two distributions.Thus, Plot 2 has a smaller standard deviation.ModelingContinuousVariablesRobb T.KoetherHomeworkReviewModelsArea AndProbabilityDensity FunctionsThe NormalDistributionsExamplesAssignmentHomework ReviewSolution(d) For Plot 1,range = 2 − (−2) = 4.x =010= 0.SSX = 4 + 4 + 1 + 1 + 0 + 0 + 1 + 1 + 4 + 4= 20.s =r209= 1.491.ModelingContinuousVariablesRobb T.KoetherHomeworkReviewModelsArea AndProbabilityDensity FunctionsThe NormalDistributionsExamplesAssignmentHomework ReviewSolution(d) For Plot 2,range = 2 − (−2) = 4.x =010= 0.SSX = 4 + 1 + 1 + 0 + 0 + 0 + 0 + 1 + 1 + 4= 12.s =r129= 1.155.ModelingContinuousVariablesRobb T.KoetherHomeworkReviewModelsArea AndProbabilityDensity FunctionsThe NormalDistributionsExamplesAssignmentModelsDefinition (Mathematical model)A mathematical model is a mathematical abstraction and,therefore, a simplification of a real situation, one that retainsthe essential features.Real situations are usually much too complicated todeal with in all their details.ModelingContinuousVariablesRobb T.KoetherHomeworkReviewModelsArea AndProbabilityDensity FunctionsThe NormalDistributionsExamplesAssignmentExampleThe “bell curve” is a model (an abstraction) of manypopulations.Real populations have all sorts of bumps and twists andirregularities.The bell curve is smooth and perfectly symmetric.In statistics, the bell curve is called the normal curve, ornormal distribution.ModelingContinuousVariablesRobb T.KoetherHomeworkReviewModelsArea AndProbabilityDensity FunctionsThe NormalDistributionsExamplesAssignmentModelsOur models will be models of distributions, presentedeither as histograms or as continuous distributions.ModelingContinuousVariablesRobb T.KoetherHomeworkReviewModelsArea AndProbabilityDensity FunctionsThe NormalDistributionsExamplesAssignmentHistograms and AreaIn a histogram, frequency is represented by area.Consider the following distribution of rainfall data.Rainfall (in) Frequency0.00 - 2.49 102.50 - 4.99 125.00 - 7.49 37.50 - 9.99 210.00 - 12.49 212.50 - 14.99 1ModelingContinuousVariablesRobb T.KoetherHomeworkReviewModelsArea AndProbabilityDensity FunctionsThe NormalDistributionsExamplesAssignmentHistograms and Area0.0 2.5 10.05.0 7.5 15.012.5630912No. of YearsRainfall (in)ModelingContinuousVariablesRobb T.KoetherHomeworkReviewModelsArea AndProbabilityDensity FunctionsThe NormalDistributionsExamplesAssignmentHistograms and AreaWhat is the total area of this histogram?ModelingContinuousVariablesRobb T.KoetherHomeworkReviewModelsArea AndProbabilityDensity FunctionsThe NormalDistributionsExamplesAssignmentHistograms and Area0.0 2.5 10.05.0 7.5 15.012.5630912No. of YearsRainfall (in)302557.552.5ModelingContinuousVariablesRobb T.KoetherHomeworkReviewModelsArea AndProbabilityDensity FunctionsThe NormalDistributionsExamplesAssignmentHistograms and AreaThe total area is25 + 30 + 7.5 + 5 + 5 + 2.5 = 75.ModelingContinuousVariablesRobb T.KoetherHomeworkReviewModelsArea AndProbabilityDensity FunctionsThe NormalDistributionsExamplesAssignmentHistograms and ProbabilityIf we select one year at random, what is the probabilitythat September rainfall was between 5 inches and 10inches?0.0 2.5 10.05.0 7.5 15.012.5630912No. of YearsRainfall (in)302557.552.5ModelingContinuousVariablesRobb T.KoetherHomeworkReviewModelsArea AndProbabilityDensity FunctionsThe NormalDistributionsExamplesAssignmentHistograms and ProbabilityWe will rescale the vertical axis so that the total areaequals 1, representing 100%.This new scale is called the density.ModelingContinuousVariablesRobb T.KoetherHomeworkReviewModelsArea AndProbabilityDensity FunctionsThe NormalDistributionsExamplesAssignmentHistograms and AreaDivide the frequencies by the original area to get thedensity.Rainfall (in) Frequency Density0.00 - 2.49 10 0.13332.50 - 4.99 12 0.16005.00 - 7.49 3 0.04007.50 - 9.99 2 0.026710.00 - 12.49 2 0.026712.50 - 14.99 1 0.0133ModelingContinuousVariablesRobb T.KoetherHomeworkReviewModelsArea AndProbabilityDensity FunctionsThe NormalDistributionsExamplesAssignmentHistograms and Area0.0 2.5 10.05.0 7.5 15.012.50.080.040.00.120.16DensityRainfall (in)ModelingContinuousVariablesRobb T.KoetherHomeworkReviewModelsArea AndProbabilityDensity FunctionsThe NormalDistributionsExamplesAssignmentHistograms and Area0.0 2.5 10.05.0 7.5 15.012.50.080.040.00.120.16DensityRainfall (in)0.33330.40000.0667 0.06670.10000.0333ModelingContinuousVariablesRobb
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