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# 12.1-12.2 - Chapter 12 Simple Linear regression and...

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Chapter 12 Simple Linear regression and Correlation So far , we discussed the methods for analyzing univariate data. Now we discuss multivariate data, obtained simultaneously on more than one variable. Bivariate Data The data 1 1 ( , )...,( , ) n n x y x y obtained on two numerical variables x and y is called a bivariate data. For, example, let x = height of a student, y = weight of a student. The data of heights and weights of all students in a class constitute a bivariate data. Scatter Plot Scatter plot is the graphyical display of a bivariate data, taking i x - values along x - axis and i y - values along the y - axis. Just plot the points 1 1 ( , )...,( , ) n n x y x y . The resulting graph is called the Scatter Plot. Usually, x = explanatory (or independent) variable y = response (or dependent) variable Often we want to predict the response variable. 1

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Examine the scatter plot for the kind of association. (i) direction (negative or positive) (ii) Strength (no, moderate, strong) (iii) From (linear or not) Example 1: The following data represents the NO x emissions of 10 engines when baseline gasoline and reformulated gasoline were used. Here x = age 1 y = Emission of NO x (baseline gasoline) 2 y = Emission of NO x (reformulated gasoline) Engine: Age: Baseline: Reformulated: 1 0 1.72 1.88 2 0 4.38 5.93 3 2 4.06 5.54 4 11 1.26 2.67 5 7 5.31 6.53 6 16 0.57 0.74 7 9 3.37 4.94 8 0 3.44 4.89 9 12 0.74 0.69 2
10 4 1.24 1.42 Age: Y-Data 18 16 14 12 10 8 6 4 2 0 7 6 5 4 3 2 1 0 Variable Baseline: Reformulated: Scatterplot of Baseline:, Reformulated: vs Age: Example 2 Consider the following data on x =BOD mass loading and y = BOD mass removal (a) Construct box plots for mean loading mass removal and comment on interesting features. 3

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(b) Construct scatter plot for the data and comment. Here, x =BOD mass loading, y = BOD mass removal x y 3 8 10 11 13 16 27 30 35 37 38 44 103 142 4 7 8 8 10 11 16 26 21 9 31 30 75 90 4 Data C2 C1 160 140 120 100 80 60 40 20 0 Boxplot of C1, C2
Example 3: The following data are on x = number of hours studied and y = the score on a test. Examine their scatter plot relationship. x 0 1 2 3 3 4 4 5 5 5 5 6 6 6 7 7 8 8 y 40 41 51 58 49 48 64 55 69 58 75 68 63 93 84 67 90 76 A way to observe such relationships is constructing a scatter plot. 5

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time score 9 8 7 6 5 4 3 2 1 0 100 90 80 70 60 50 40 Scatterplot of score vs time 12.5 Correlation Coefficient 6
Scatter plot gives only a visual impression on the relationship between x and y ; some times eyes may be fooled. There is a need for a precise statement, and it is given by Karl Pearson’s Correlation Coefficient. This reassures the strengthening of linear association between x and y . 7

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Definition. The correlation measures the direction and the strength of the linear relationship between x and y . It is given by 1 1 1 n i i i x y x x y y r n s s = - - = - , 8
where x , and x s are the sample mean and sample standard deviation of 1 ,..., n x x . Similarly for y and y s . Note r = r ( x

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12.1-12.2 - Chapter 12 Simple Linear regression and...

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