Class notes for Chapter#1.pdf - Chapter 1 Linear Regression with One Predictor Variable 1.1 Statistical Relation Between Two Variables To motivate

Class notes for Chapter#1.pdf - Chapter 1 Linear Regression...

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Chapter 1. Linear Regression with One Predictor Variable 1.1 Statistical Relation Between Two Variables To motivate statistical relationships, let us consider a mathematical relation between two mathematical variables x and y . This may be represented by a functional rela- tion; y = f ( x ) , (1) which says that given a value of x, there is a unique value of y, which can be exactly determined. 1 For example, the relation between the number of hours( x ) driven on a car and distance ( y ) travelled may be given by y = cx, where c is the constant speed. There are many examples in physical and other sciences of such relations, known as the deterministic or exact relationship. To define a statistical relationship, we re- place the mathematical variables by ran- dom variables, X and Y and add a random component of error representing devia- tion from the true relation is given by y = f ( x ) + (2) 2
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Here ( x, y ) represent a typical value of the bi- variate random variable ( X, Y ) . Such a relation is also known as stochas- tic relation and models the random phe- nomenon where (i) there is tendency of Y values to vary around a smooth function and (ii) there is a random scatter of points around this systematic component. Figure 1.1 presents the plot of heights and weights of 23 students enrolled in my 2001 class of STAT360 (for the data given in Table 1.1). 3 This graph shows the tendency of the data to vary around a straight line. This ten- dency of the variation in weights as func- tion of height is called linear trend. Since the points do not fall on a straight line, it may be suitable to use a statistical rela- tionship, i.e. y = β 0 + β 1 x + where β 0 and β 1 are unknown constants, x represents height and y represents weight, and represents a random error. The subject matter of this course is the study of such relationships. 4
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Figure 1.1 Scatter Plot of Height-Weight Data of STAT360 2001 Class 5 Table 1.1 Heights and Weights of 23 Students in STAT 360 Class of 2001 Student ID Height(Cms.) Weight(Kgs.) 4126548 183.00 77.09 4281675 177.80 90.70 4100212 172.72 81.63 4411919 167.64 49.88 5936748 162.56 45.35 5919460 162.56 54.42 5945267 172.72 72.56 4276051 177.80 74.83 4084489 172.72 54.42 4139615 185.42 92.97 5928281 180.34 81.63 5922763 172.72 80.72 3630137 180.34 70.29 4751612 158.00 55.00 4767098 163.00 50.00 4767209 158.00 42.00 4766733 182.00 72.00 4766164 166.00 60.00 4763661 168.00 62.00 4766970 163.00 55.00 4763734 170.00 65.00 3952312 172.72 95.23 5928389 162.56 72.56 6
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1.2 Regression Models Terminology: Regression The conditional expectation given by m ( x ) = E ( Y | X = x ) in a bivariate setting is called regression of Y on X . The term regression was used by Sir Francis Galton (1822-1911) in study- ing the height of the offsprings as a func- tion of the heights of their parents in a paper entitled ”Regression towards medi- ocrity in hereditary stature” ( Nature , vol. 15, pp.507-510). 7 In this paper Galton reported on his discov- ery that ”the offsprings did not resemble their parents in size but tend to be always more mediocre [i.e. more average] than they - to be smaller than the parents if par- ents were large; to be larger than parents if they were very small...” Thus the random variable Y may be as- sumed to vary around its mean m ( x ) as a function of X , and denoting the random deviation Y m ( x ) by , we can write Y = m ( x ) + (3) Note that the probability distribution of
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