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# FINITE MATHEMATICS MATH 170

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This 63 page Class Notes was uploaded by Cassidy Grimes on Monday October 26, 2015. The Class Notes belongs to MATH 170 at University of South Carolina - Columbia taught by M. Gamel in Fall. Since its upload, it has received 40 views. For similar materials see /class/229519/math-170-university-of-south-carolina-columbia in Mathematics (M) at University of South Carolina - Columbia.

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Date Created: 10/26/15

Matthew Gamel September 2 2008 Matthew Gamel mm r gm Matthew Gamel Linear Re An JEJEmntentafy HUUMSLf amm Consider the table of data Are there any patterns in the data set above Matthew Gamel may An JEJEmnrentary HUles Lfrainm Consider the table of data Are there any patterns in the data set above Yes there are Matthew Gamel may An El remeni cary Ht ws tramm Consider the table of data Are there any patterns in the data set above Yes there are The data in the table all He exactly on a line given by y 3m 1 Matthew Gamel mwzv An Elementary lillws tiramion Consider the table of data Are there any patterns in the data set above Yes there are The data in the table all lie exactly on a line given by y 3x l 1 However life is normally not this pleasent in practice Matthew Gamel imwzv x H 1 15 25 935 H 08 09 085 093 097 Matthew Gamel Linear Regression An Ei ennieni cafy iiiiiiiyis titamion Now consider the following table of data ix H1 15i2 i25i3 i igmH08i09i085i093i097i What makes this set of data so different Matthew Gamel igmrzv An El ennieni cafy lillws tfamion Now consider the following table of data lz H1 15l2 l25l3 l lgmll08l09l085l093l097l What makes this set of data so different These points do not all lie exactly on a line Why Matthew Gamel igmrzv A10 Elementary lilJllws Lframion Now consider the following table of data 1 H 1 15 25 l g H 08 l 09 l 085 l 093 l 097 l What makes this set of data so different These points do not all lie exactly on a line Why Because from x 1 to z 15 the values increase by 01 but from x 15 to z 2 they decrease by 005 Matthew Gamel igmrzv Here IS the same set of data WIth a best fit line through the pomts y 0074I 0742 Conclusion and Recap Our goal is to attempt to determine how to find the line that best represents a given set of data Matthew Gamel mer 2 Conclusion and Recap Our goal is to attempt to determine how to find the line that best represents a given set of data Of course we will have to specify what we mean by best represents39 Matthew Gamel mer 2 What is Linear Regression The example that we gave does not really drive the point home Matthew Gamel Linear R The example that we gave does not really drive the point home Typically in practice one might in principle collect thousands of data points and will want to find a line that best fits the data Matthew 5 am el n Gun M and man What is Linear Regression The example that we gave does not really drive the point home Typically In practice one might in principle collect thousands of data points and Will want to find a line that best fits the data What is Linear Regression We need to precisely specify what we mean by best fits39 Matthew Gamel Linear R What is Linear Regression We need to precisely specify what we mean by best fits I3 i5 39 39 i 3x r39 Matthew Gamel Linear Regression Why Regression The Basic Idea Residuals and Error Conclusion and Recap What is Linear Regression We need to precisely specify what we mean by best fits We want to minimize the square of the distance from each point to the line Matthew Gamel Linear Regression Least Squares Regression Because we minimize the square of the distance this is called linear least squares regression Matthew Gamel Linear R Conclusion and Recap a R gm m Because we minimize the square of the distance this is called linear least squares regression One might reasonably ask why we minimize the distance of the squares instead of the actual distance Matthew Gamel imam ii x Because we minimize the square of the distance this is called linear least squares regression One might reasonably ask why we minimize the distance of the squares instead of the actual distance Unfortunately the answer is rather technical Matthew Gamel lumer ii x Because we minimize the square of the distance this is called linear least squares regression One might reasonably ask why we minimize the distance of the squares instead of the actual distance Unfortunately the answer is rather technical The answer lurkes within the proof of the theorem that we will shortly state and involves some elementary vector calculus Matthew Gamel lumer ii x How Do We Do It We are almost ready to state a theorem that tells us how to do this Matthew Gamel Linear R Conclusion and Recap lHl Wi De We lD lit We are almost ready to state a theorem that tells us how to do this But first recall that if m1 mm is a set of n points then it39s arithmetic mean or average is given by Matthew Gamel iner ii x 19 c Residuals and Conclusion and Recap lit lHl Wi Den iiiill We are almost ready to state a theorem that tells us how to do this But first recall that if m1 mm is a set of n points then it39s arithmetic mean or average is given by 7 1n 12mn gasp Matthew Gamel may 61 Ie Idea esiJuals and Error Conclusion and Recap lHl Wr Die We lit We are almost ready to state a theorem that tells us how to do this But first recall that if m1 mm is a set of n points then it39s arithmetic mean or average is given by 7 1n 12mn 3072130 71 The following theorem then tells us how to compute the regression line Matthew Gamel imwzv ii r nd Er an d r mp HJ IM Die 33er Hit Theorem Let 17y1zmyn be a data set ofn points Then the regression line that best fits the data is given by x ax b Where Matthew Gamel mwzv r Tre Idea Residuals and Error Conclusion and Recap Hit HJ IM Die Wife Theorem Let 17y1zmyn be a data set ofn points Then the regression line that best fits the data is given by x ax b Where a n and 2 7 72 E j1mj 7m Matthew Gamel may r 19 c Residuals and Conclusion and Recap HJ IM Den Wife Hit Theorem Let 17y1zmyn be a data set ofn points Then the regression line that best fits the data is given by x ax b Where 7 2 7 and b 921j 21jyj Matthew Gamel may Gm How Do We Do It This looks rather complicated but it really is not Matthew Gamel Linear R Conclusion and Recap lHl Wi Do We lD lit This looks rather complicated but it really is not There are fundamentally only four things that need to be computed Matthew Gamel igmwzv ii x Conclusion and Recap lHl Wi Dion Till73 lD lit This looks rather complicated but it really is not There are fundamentally only four things that need to be computed lgt The averages E and y Matthew Gamel igmwzv ii x ml Er an d r mp lHl Wr Die Till73 lit This looks rather complicated but it really is not There are fundamentally only four things that need to be computed lgt The averages E and y 71 D The quantity Sm ijyj j1 Matthew Gamel imwzv ii r lHl Wr Dion Till73 lD lit This looks rather complicated but it really is not There are fundamentally only four things that need to be computed lgt The averages E and y 71 D The quantity Sm ijyj j1 V L rgt The quantity S j1 Matthew 5 am el lHl Wr Dion lily73 lD lit This looks rather complicated but it really is not There are fundamentally only four things that need to be computed lgt The averages E and y 71 D The quantity Sm ijyj j1 V L rgt The quantity S j1 Once you have computed these things it is then only a matter of plugging them into the formula Matthew Gamel lunar Example 1 Let39s revisit the example that we first considered Matthew Gamel Linear R Let39s revisit the example that we first considered Suppose that we want to compute the least squares regression line for the following set of data Matthew 5 am el begi midle 1 Let39s revisit the example that we first considered Suppose that we want to compute the least squares regression line for the following set of data There are 5 data points ie n 5 Matthew Gamel mew Let39s revisit the example that we first considered Suppose that we want to compute the least squares regression line for the following set of data There are 5 data points ie n 5 We now proceed to make the necessary computations Matthew 5 am el lih v P m The Basic Idea Example 1 E 11522531052 Matthew Gamel Linear Re Example 1 11522531052 l 5 08 09 085 093 097 4455 089 Matthew Gamel Linear Regression aslc Idea nd Error m j g 1 11522531052 y l 5 08 09 085 093 097 4455 089 5 SE 5 12 15 22 25 32 225 j1 Matthew Gamel mwzv Emma 1 11522531052 y l 5 08 09 085 093 097 4455 089 108 1509 2085 25093 3097 9085 Matthew Gamel mwzv Example 1 The formula tells us that the line is y ax l b where Matthew Gamel Linear R Emm mlpll g ll The formula tells us that the line is y ax l b where Sm 7 nfy i 9085 7 5 2089 i 0185 0074 Sm 7 552 225 7 522 a Matthew Gamel may aslc Idea nd Error The formula tells us that the line is y ax l b where Sm 7 55 g 9085 7 52089 0185 7 7 0074 SE 7 mi 225 7 522 25 351 7 ES 089225 7 522 7 2 b Sm 7 552 225 7 522 0742 9085 7 1855 7 25 Matthew Gamel mar aslc Idea nd Error The formula tells us that the line is y ax l b where Sm 7 71 g 9085 7 5 2089 0185 a i i 7 7 7 0074 SE 7 mg 225 522 25 b 0742 7 ySm 755M 7 089225 7 522 7 29085 7 1855 7 5m 7 nEZ 225 7 522 25 The regression line is then given by y 0074z l 0742 Matthew Gamel mew Conclusion and Recap WainM annariks When using the formula for the regression line note that the denominators for a and b are the same There is therefore no need to compute it twice Matthew Gamel Jumer ii x Conclusion and Recap lHlellplFiull annarlks When using the formula for the regression line note that the denominators for a and b are the same There is therefore no need to compute it twice Specifically the denominator is given by Matthew Gamel lumer ii x T19 c Residuals and Conclusion and Recap lHksllp ull ieinnanks When using the formula for the regression line note that the denominators for a and b are the same There is therefore no need to compute it twice Specifically the denominator is given by n EmginE2SminE2 11 Matthew Gamel may 61 How Good is Our Approximation Matthew Gamel Linear R R Conch on Concluding Remarks We begin with a table of data Matthew Gamel Linear R an d R ecap Cendiyiding P3 We begin with a table of data Our goal is to find a linear function y am i b that is the line which best fits the data in a least squares sense Matthew 5 am el an d R ecap Conduding P3 We begin with a table of data Our goal is to find a linear function y am i b that is the line which best fits the data in a least squares sense If the data precisely represents data on a line there is nothing hard to do but compute the equation of the line Matthew 5 am el an d R ecap Concluding P3 We begin with a table of data Our goal is to find a linear function y am i b that is the line which best fits the data in a least squares sense If the data precisely represents data on a line there is nothing hard to do but compute the equation of the line If the data does not lie on a line we choose a and bin a way that the square of the distance from each point to the line is minimal Matthew 5 am el w my Err Conch on and Recap Concluding Remarks We must compute the following quantities Matthew Gamel Linear R w my Err Conch on and Recap Concluding Remarks We must compute the following quantities gt The averages E and y Matthew Gamel Linear R Concluding Remarks We must compute the following quantities gt The averages E and y gt The quantity S 21 Matthew Gamel Linear R nd Err Conch on and Recap Pie hiding mavhg We must compute the following quantities gt The averages E and y gt The quantity Sm 21 gt The quantity Sm 21 zjyj Matthew Gamel mew r w Idea Residuals and Error Conclusion and Recap Pie nneawhg Jy rd ng We must compute the following quantities gt The averages E and y gt The quantity Sm 21 gt The quantity Sm 21 zjyj D The denominator D 21 7 7amp2 Sm 7 7amp2 Matthew Gamel may r an d R ecap Condwd ng TR We must compute the following quantities gt The averages E and y gt The quantity S 21 gt The quantity Sm 21 zjyj D The denominator D 21 7 7amp2 Sm 7 7amp2 lgt The coeficients a and b Smy 71 y a b m a my D D Matthew 5 am el

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