CHAPTER 3 INTRODUCTORY LINEAR REGRESSION

Introduction  Linear regression is a study on the linear relationship between two variables. This is done by fitting a linear equation to the observed data.  The linear equation is then used to predict values for the data.  In a simple linear relationship, only two variables are involved: a)X is the independent variable - the variables has been controlled b)Y is the dependent variable - the response variables. In other word, the value of y depends on the value of x.

Example A nutritionist studying weight loss programs might wants to find out if reducing intake of carbohydrate can help a person reduce weight. a)X is the carbohydrate intake (independent variable). b)Y is the weight (dependent variable). An entrepreneur might want to know whether increasing the cost of packaging his new product will have an effect on the sales volume. a)X is cost b)Y is sales volume

Scatter plots  A scatter plot is essentially a plot between the pair of (x,y) values.  The purpose of constructing the plot is to examine the relationship between the two variables.

(a) Positive linear relationship

 A linear regression equation is a mathematical equation that can be used to predict the values of one dependent variable from known values of an independent variable.  This equation represents a straight line so it is of the form, y=mx+c, where m is the slope and c is the y-intercept.  The true regression line or the probabilistic model is given by,

 We will call this model the simple linear regression model because it has only one independent variable.  This regression line is estimated from the data collected by fitting a straight line to the data set and getting the equation of the straight line,

Least Squares Method  The least squared method is the commonly method used for estimating the regression coefficient,.  The straight line fitted to the data set is the line. Example

 The analysis of variance (ANOVA) method is an approach to test the significance of the regression. We can arrange the test procedure using this approach in an ANOVA table as shown below  The test hypotheses are  We will reject if at α level of significance. Then we conclude there exist a linear relationship between the two variable being investigated.

Correlation Correlation measures the strength of a linear relationship between the two variables. One numerical measure is the Pearson product moment correlation coefficient, r. Properties of r:   Values of r close to 1 implies there is a strong positive linear relationship between x and y.  Values of r close to -1 implies there is a strong negative linear relationship between x and y.  Values of r close to 0 implies little or no linear relationship between x and y.

Before After

Suppose you wish to investigate the relationship between the numbers of hours student’s spent studying for an examination and the mark they achieved. StudentsABCDEFGH numbers of hours (x) Final marks ( (y) Numbers of hours student’s spent studying for an examination ( x – Independent variable ) the mark (y) they achieved. ( y – Dependent variable ) will cause

Strong Linear positive correlation 89.26% of variation in marked achieved is due to variation in numbers oh hours student’s spent studying

 This chapter introduces important methods (regression) for making inferences about a relationship between two variables and describing such a relationship with an equation that can be used for predicting value of one variable given the value of the other variable.