Modeling Continuous Variables Lecture 19 Section 6 1 6 3 1 Fri Feb 24 2006 Models Mathematical model An abstraction and therefore a simplification of a real situation one that retains the essential features Real situations are usually much to complicated to deal with in all their details Examples Economic models treat money as a continuous quantity even though it is discrete This is an abstraction that is incorporated into the model to make it simpler The bell curve is a model an abstraction of many populations Real populations have all sorts of bumps and twists The bell curve is smooth and perfectly symmetric Models No mathematical model is perfect A mathematical model is useful and powerful to the extent that it is a faithful representation of reality Conversely to the extent that is it not faithful to reality it can lead to false conclusions about the situation that it is supposed to model Example of a Model Use a random number generator to simulate how a pair of rolled dice will land The possible totals range from 2 to 12 Using the TI 83 which is a correct model Enter randInt 2 12 that is get a random number from 2 to 12 or Enter 2 randInt 1 6 that is double a random number from 1 to 6 or Enter randInt 1 6 randInt 1 6 that is add two random numbers from 1 to 6 Histograms and Area If a histogram is drawn appropriately then frequency is represented by area Consider the following histogram of test scores Grade Frequenc 60 69 70 79 80 89 90 99 y 3 8 9 5 Histograms and Area Frequency 10 8 6 4 2 Grade 0 60 70 80 90 100 Histograms and Area In the histogram we may replace the frequency with the proportion of the total Grade Frequency Proportion 60 69 3 0 12 70 79 8 0 32 80 89 9 0 36 90 99 5 0 20 Histograms and Area Proportion 0 40 0 30 0 20 0 10 Grade 0 60 70 80 90 100 Histograms and Area Proportion 0 40 0 30 0 20 0 10 Grade 0 60 70 80 90 100 Histograms and Area Furthermore we may divide the proportions by the width of the classes to get the density Grade Frequency Proportion Density 60 69 3 0 12 0 012 70 79 8 0 32 0 032 80 89 9 0 36 0 036 90 99 5 0 20 0 020 Histograms and Area Density 0 040 0 030 0 020 0 010 Grade 0 60 70 80 90 100 Histograms and Area The final histogram has the special property that the proportion can be found by computing the area of the rectangle The vertical scale has been adjusted so that the total area is 1 or 100 For example what proportion of the grades are less than 80 Compute 10 0 012 10 0 032 0 12 0 32 0 44 44 Density Functions This is the fundamental property that connects the graph of a continuous model to the population that it represents namely The area under the graph between two numbers a and b on the x axis represents the proportion of the population that lies between a and b AREA PROPORTION Density Functions The area under the curve between a and b is the proportion of the values of x that lie between a and b x a b Density Functions The area under the curve between a and b is the proportion of the values of x that lie between a and b x a b Density Functions The area under the curve between a and b is the proportion of the values of x that lie between a and b x a b Area Proportion Density Functions A consequence of this is that the total area under the curve must be 1 representing a proportion of 100 x a b Density Functions A consequence of this is that the total area under the curve must be 1 representing a proportion of 100 100 a x b The Normal Distribution Normal distribution The statistician s name for the bell curve It is a density function in the shape of a bell sort of Symmetric Unimodal Extends over the entire real line no endpoints Main part lies within 3 of the mean The Normal Distribution The curve has a bell shape with infinitely long tails in both directions The Normal Distribution The mean is located in the center at the peak The Normal Distribution The width of the main part of the curve is 6 standard deviations wide 3 standard deviations each way from the mean 3 3 The Normal Distribution The area under the entire curve is 1 The area outside of 3 st dev is approx 0 0027 Area 1 3 3 The Normal Distribution The normal distribution with mean and standard deviation is denoted N For example if X is a variable whose distribution is normal with mean 30 and standard deviation 5 then we say that X is N 30 5 The Normal Distribution If X is N 30 5 then the distribution of X looks like this 15 30 45 Some Normal Distributions N 3 1 0 1 2 3 4 5 6 7 8 Some Normal Distributions N 5 1 N 3 1 0 1 2 3 4 5 6 7 8 Some Normal Distributions N 2 N 5 1 N 3 1 0 1 2 3 4 5 6 7 8 Some Normal Distributions N 2 N 3 1 N 5 1 N 3 1 0 1 2 3 4 5 6 7 8 Bag A vs Bag B Suppose we have two bags Bag A and Bag B Each bag contains millions of vouchers In Bag A the values of the vouchers have distribution N 50 10 Normal with 50 and 10 In Bag B the values of the vouchers have distribution N 80 15 Normal with 80 and 15 Bag A vs Bag B H0 Bag A H1 Bag B 30 40 50 60 70 80 90 100 110 Bag A vs Bag B We are presented with one of the bags We select one H Bag A voucher at random from that bag 0 H1 Bag B 30 40 50 60 70 80 90 100 110 Bag A vs Bag B If its value is less than or equal to 65 then we will decide that it was from Bag A H Bag A 0 H1 Bag B 30 40 50 60 65 70 80 90 100 110 Bag A vs Bag B If its value is less than or equal to 65 then we will decide that it was from Bag A H Bag A 0 H1 Bag B 30 40 50 Acceptance Region 60 65 70 80 90 100 110 Bag A vs Bag B If its value is less than or equal to 65 then we will decide that it was from Bag A H Bag A 0 H1 Bag B 30 40 50 Acceptance Region 60 65 70 80 90 100 Rejection Region 110 …
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