CHAPTER 10 Correlation and Regression Objectives n Draw

CHAPTER 10 Correlation and Regression (Objectives) n Draw a scatter plot for a set of ordered pairs. n Compute the correlation coefficient. n Test the hypothesis H 0: 0. (Will be done later) n Compute the equation of the regression line. 2/13/2022 © Kasturiarachi 1

Statistical Methods n Correlation is a statistical method used to determine whether a linear relationship between variables exists. n Regression is a statistical method used to describe the nature of the relationship between variables—that is, positive or negative, linear or nonlinear. 2/13/2022 © Kasturiarachi 2

Statistical Questions 1. Are two or more variables related? 2. If so, what is the strength of the relationship? 3. What type or relationship exists? 4. What kind of predictions can be made n A correlation coefficient is a measure of how variables are related. n In a simple relationship, there are only two types of variables under study. 2/13/2022 © Kasturiarachi from the relationship? 3

Scatter Plots n A scatter plot is a graph of the ordered pairs (x, y) of numbers consisting of the independent variable, x, and the dependent variable, y. n A scatter plot is a visual way to describe the nature of the relationship between the independent and dependent variables. 2/13/2022 © Kasturiarachi 4

(10. 2): Correlation Coefficient n The correlation coefficient computed from the sample data measures the strength and direction of a linear relationship between two variables. n The symbol for the sample correlation coefficient is r. n The symbol for the population correlation coefficient is . 2/13/2022 © Kasturiarachi 5

Correlation Coefficient (cont’d. ) n The range of the correlation coefficient is from 1 to 1. n If there is a strong positive linear relationship between the variables, the value of r will be close to 1. n If there is a strong negative linear relationship between the variables, the value of r will be close to 1. 2/13/2022 © Kasturiarachi 6

Correlation Coefficient (cont’d. ) n When there is no linear relationship between the variables or only a weak relationship, the value of r will be close to 0. 1 Strong negative linear relationship No linear relationship 0 2/13/2022 © Kasturiarachi 1 Strong positive linear relationship 7

Formula for the Correlation Coefficient r n where n is the number of data pairs. 2/13/2022 © Kasturiarachi 8

Possible Relationships Between Variables n There is a direct cause-and-effect relationship between the variables: that is, x causes y. n There is a reverse cause-and-effect relationship between the variables: that is, y causes x. n The relationship between the variable may be caused by a third variable: that is, y may appear to cause x but in reality z causes x. 2/13/2022 © Kasturiarachi 9

(10. 3): Regression Line n If the value of the correlation coefficient is significant, the next step is to determine the equation of the regression line which is the data’s line of best fit. n Best fit means that the sum of the squares of the vertical distance from each point to the line is at a minimum. 2/13/2022 © Kasturiarachi 10

Scatter Plot with Three Lines 2/13/2022 © Kasturiarachi 11

A Linear Relation 2/13/2022 © Kasturiarachi 12

Equation of a Line n In algebra, the equation of a line is usually given as , where m is the slope of the line and b is the y intercept. n In statistics, the equation of the regression line is written as , where b is the slope of the line and a is the y' intercept. 2/13/2022 © Kasturiarachi 13

Regression Line n Formulas for the regression line : where a is the y' intercept and b is the slope of the line. 2/13/2022 © Kasturiarachi 14

Regression Line (Easy Formula) n The formula for the regression line: where slope and intercept 2/13/2022 © Kasturiarachi 15