1. STATISTICS 12.0 Correlation and Linear Regression “Correlation and Linear Regression -”Causal Forecasting Method

2. Correlation and Linear Regression –This chapter focuses on describing how two variables relate to one another. –Using correlation and linear regression: a) Determine whether a relationship does exist between the variables b) Describe the nature of this relationship in mathematical terms 12.0 Correlation and Linear Regression

3. Independent vs Dependent Variables Data For Statistics Exam Hours Studied Exam Grade 386 595 492 483 278 382 12.0 Correlation and Linear Regression

4. Independent vs Dependent Variables –The hours studied variable is considered the independent variable (x), exam grade is considered the dependent variable (y) –The independent variable (x) causes variation in the dependent variable (y) –The data is considered ordered pair of (x,y) values. Eg. (3,86) and (5,95) 12.0 Correlation and Linear Regression

5. CORRELATION 1)Correlation measures both the strength and direction of the relationship of x and y. 2) The convention is to place the x variable on the horizontal axis and the y variable on the vertical axis. 3)There are 4 different types of correlation in a series of scatter plots: a)Positive Linear Correlation b)Negative Linear Correlation c)No Correlation d)Nonlinear Correlation 12.0 Correlation and Linear Regression

6. CORRELATION 1)Correlation Coefficient, r indicates with both strength and direction of the relationship 2) Values of r range between -1.0 and +1.0 3)We can calculate the actual correlation coefficient using: r = n ∑ xy – (∑x)(∑y) √ [n∑x² -(∑x)²][n∑y² - (∑y)²] 12.0 Correlation and Linear Regression

7. LINEAR REGRESSION 1)The technique of simple regression enables us to describe a straight line that best fits a series of ordered pairs (x,y) 2) The equation for a straight line, known as a linear equation, take from: ŷ = a + bx x = the independent variable ŷ = the predicted value of y, given a value of x a = the y-intercept for the straight line b = the slope of the straight line 12.0 Correlation and Linear Regression

8. LINEAR REGRESSION 3)The least squares method finds the linear equation to calculate a, the y-intercept and b, the slope: b = n ∑ xy – (∑x)(∑y) a = y - b x n∑x² -(∑x)² 12.0 Correlation and Linear Regression x = the average value of y- dependent y = the average value of x-independent

9. LINEAR REGRESSION 4) Standard error of estimate s e is to estimate how accurate the estimated/predicted number variables: s e = ∑y² - a ∑y - b∑ xy n-2 5) The coefficient of determination r² represent the percentage of variation y that is explained by the regression line 12.0 Correlation and Linear Regression √

10. Exercises The price of 2 nd hand car per mileage Mileage (K) Price (K) 21.8$16 34.0$11.5 41.7$13.4 53.5$14.8 65.8$10.5 72.1$12.3 76.5$8.20 84.7$9.50 a) Calculate the correlation coefficient. b) What is the regression line that best fits the data? (Please draw to support your answer) c) What is the predicted price of a 2 nd hand car with the mileage of 10.7K? d) Calculate the standard error of estimate the regression line and the coefficient of determinance.