Modeling Continuous Variables Robb T Koether Homework Review Models Area And Probability Modeling Continuous Variables Lecture 19 Sections 6 1 6 3 1 Density Functions The Normal Distributions Robb T Koether Examples Assignment Hampden Sydney College Tue Feb 17 2009 Outline Modeling Continuous Variables Robb T Koether Homework Review 1 Homework Review 2 Models 3 Area And Probability Density Functions 4 The Normal Distributions Examples 5 Assignment Models Area And Probability Density Functions The Normal Distributions Examples Assignment Homework Review Modeling Continuous Variables Robb T Koether Homework Review Models Exercise 5 20 page 336 Consider the following two frequency plots based on two sets of data with 10 observations Frequency Plot 1 Frequency Plot 2 Area And Probability Density Functions The Normal Distributions Examples Assignment 2 1 0 1 2 2 1 0 1 2 Note that Plot 1 can be converted to Plot 2 by moving the two blue X s Homework Review Modeling Continuous Variables Robb T Koether Homework Review Models Area And Probability Density Functions The Normal Distributions Examples Assignment Exercise 5 20 page 336 a Do you think the range for Plot 1 is smaller than equal to or larger than the range for Plot 2 b Do you think the mean for Plot 1 is smaller than equal to or larger than the mean for Plot 2 c Do you think the standard deviation for Plot 1 is smaller than equal to or larger than the standard deviation for Plot 2 Explain d Check your answers to parts a b and c by computing the range mean and standard deviation for each set of data Homework Review Modeling Continuous Variables Robb T Koether Homework Review Models Area And Probability Solution a The ranges are clearly equal b Both distributions are symmetric so the means are in the center at 0 Thus they are equal Density Functions The Normal Distributions Examples Assignment c The two blue points in Plot 2 are closer to the mean than they are in Plot 1 so they have smaller deviations All other points are the same between the two distributions Thus Plot 2 has a smaller standard deviation Homework Review Modeling Continuous Variables Robb T Koether Homework Review Models Area And Probability Density Functions The Normal Distributions Examples Assignment Solution d For Plot 1 2 2 4 0 x 0 10 SSX 4 4 1 1 0 0 1 1 4 4 range r 20 20 s 9 1 491 Homework Review Modeling Continuous Variables Robb T Koether Homework Review Models Area And Probability Density Functions The Normal Distributions Examples Assignment Solution d For Plot 2 2 2 4 0 x 0 10 SSX 4 1 1 0 0 0 0 1 1 4 range r 12 12 s 9 1 155 Models Modeling Continuous Variables Robb T Koether Homework Review Models Area And Probability Density Functions The Normal Distributions Examples Assignment Definition Mathematical model A mathematical model is a mathematical abstraction and therefore a simplification of a real situation one that retains the essential features Real situations are usually much too complicated to deal with in all their details Example Modeling Continuous Variables Robb T Koether Homework Review Models Area And Probability Density Functions The Normal Distributions Examples Assignment The bell curve is a model an abstraction of many populations Real populations have all sorts of bumps and twists and irregularities The bell curve is smooth and perfectly symmetric In statistics the bell curve is called the normal curve or normal distribution Models Modeling Continuous Variables Robb T Koether Homework Review Models Area And Probability Density Functions The Normal Distributions Examples Assignment Our models will be models of distributions presented either as histograms or as continuous distributions Histograms and Area Modeling Continuous Variables Robb T Koether Homework Review Models Area And Probability Density Functions The Normal Distributions Examples Assignment In a histogram frequency is represented by area Consider the following distribution of rainfall data Rainfall in 0 00 2 49 2 50 4 99 5 00 7 49 7 50 9 99 10 00 12 49 12 50 14 99 Frequency 10 12 3 2 2 1 Histograms and Area Modeling Continuous Variables 12 Robb T Koether Homework Review Area And Probability Density Functions The Normal Distributions 9 No of Years Models 6 Examples Assignment 3 0 0 0 2 5 5 0 7 5 10 0 Rainfall in 12 5 15 0 Histograms and Area Modeling Continuous Variables Robb T Koether Homework Review Models Area And Probability Density Functions The Normal Distributions Examples Assignment What is the total area of this histogram Histograms and Area Modeling Continuous Variables 12 Robb T Koether Homework Review Area And Probability Density Functions The Normal Distributions 9 No of Years Models 6 30 25 Examples Assignment 3 7 5 5 5 0 0 0 2 5 5 0 7 5 10 0 Rainfall in 2 5 12 5 15 0 Histograms and Area Modeling Continuous Variables Robb T Koether Homework Review Models The total area is Area And Probability Density Functions The Normal Distributions Examples Assignment 25 30 7 5 5 5 2 5 75 Histograms and Probability Modeling Continuous Variables Robb T Koether If we select one year at random what is the probability that September rainfall was between 5 inches and 10 inches Homework Review 12 Models Area And Probability The Normal Distributions Examples Assignment 9 No of Years Density Functions 6 30 25 3 7 5 5 5 0 0 0 2 5 5 0 7 5 10 0 Rainfall in 2 5 12 5 15 0 Histograms and Probability Modeling Continuous Variables Robb T Koether Homework Review Models Area And Probability Density Functions The Normal Distributions Examples Assignment We will rescale the vertical axis so that the total area equals 1 representing 100 This new scale is called the density Histograms and Area Modeling Continuous Variables Robb T Koether Homework Review Models Area And Probability Density Functions The Normal Distributions Examples Assignment Divide the frequencies by the original area to get the density Rainfall in 0 00 2 49 2 50 4 99 5 00 7 49 7 50 9 99 10 00 12 49 12 50 14 99 Frequency 10 12 3 2 2 1 Density 0 1333 0 1600 0 0400 0 0267 0 0267 0 0133 Histograms and Area Modeling Continuous Variables 0 16 Robb T Koether Homework Review 0 12 Area And Probability Density Functions The Normal Distributions Density Models 0 08 Examples Assignment 0 04 0 0 0 0 2 5 5 0 7 5 10 0 Rainfall in 12 5 15 0 Histograms and Area Modeling Continuous Variables 0 16 Robb T Koether Homework Review 0 12 Area And Probability Density Functions The Normal Distributions Density Models 0 4000 0 08 0 3333 Examples
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