Simple Linear Regression
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This document provides information about regression analysis and linear regression. It defines regression analysis as using relationships between quantitative variables to predict a dependent variable from independent variables. Linear regression finds the best fitting straight line relationship between variables. The simple linear regression equation is given as Y = a + bX, where a and b are estimated parameters calculated from sample data. An example is worked through, showing how to calculate the regression equation from data, graph the relationship, and use the equation to estimate values.
Simple Linear Regression
- 2. Regression Analysis Is a statistical method which makes use of the relationships between two or more quantitative variables so that on variable, called the dependent or response variable can be predicted with the knowledge of the values of the other variable, called independent variable. A regression equation is a mathematical equation that allow us to predict values of one dependent variable from known values of one or more independent variables.
- 3. The regression equation is usually expressed mathematically by a straight line equation called linear regression line or simple linear regression. This line will give the best fit to the relationship of the variables X and Y. The lie which “best fit” is that line such that when the differences between the actual values of Y and the predicted values of Y for each X are squared and summed, it will result a minimum.
- 4. The simple linear regression line is given by the equation Y = a + bX where Y is the predicted dependent variable, X is the independent variable, a and b are the estimates of the parameters of regression which are calculated from the available sample values as follows: N ( XY) – ( X) ( Y) b= N( X2 ) – ( X)2 and a = y – bx where y and x the means of the sample values of X and Y.
- 5. Example: Consider the following data: X 2 4 6 8 10 12 Y 6 7 8 9 10 11 a. Find the equation of regression line b. Sketch the graph on a scatter diagram c. Find the point estimate of Y when X = 15
- 6. Solution (a): X Y X2 Y2 XY 2 4 6 7 4 16 36 49 12 28 6 10 8 9 10 36 64 100 64 81 100 48 72 100 12 11 144 121 132 X2 = 364 Y2 = XY = 392 8 X = 42 X=7 Y = 51 Y= 8.5 451
- 7. Then: 6(392) – 42(51) b= 210 = 6(364) – 422 = 0.5. 420 Thus, a = 8.5 – 0.5(7) = 5. Therefore, the equation of the regression line is Y = 5 + 0.5X.
- 8. Solution (b): Y-Values 14 12 12 10 10 8 8 6 Y-Values 6 4 4 2 2 0 2 4 6 8 10 12 Solution (c) : Substitute x = 15 to the equation Y = 5 + 0.5X. Thus, Y = 5 + 0.5(15) = 12.5
- 9. Problem: A study was made by a businessman to determine the relation between advertising cost and sales. The following data on 12 commodities were recorded: Advertisin 30 g cost (thousand) 15 400 Sales (thousand) 320 350 490 500 500 530 24 37 42 45 48 40 20 25 20 385 450 390 365 470 Find the estimated regression line and estimate the sales when X = 43. 35