1 Dr. Jerrell T. Stracener EMIS 7370 STAT 5340 Probability and Statistics for Scientists and Engineers Department of Engineering Management, Information and Systems Simple Linear Regression

2 Regression Analysis is a statistical analysis used to investigate and draw conclusions about functional relationships existing between a dependent variable and one or more independent variables Simple Linear Regression

3 Multiple linear regression model where Y is the response (or dependent) variable X 1, X 2,..., X k are the independent variables  0,  1,  2,..., k are the unknown parameters and  is the experimental error Multiple Linear Regression Model

4 Simple linear regression model where Y is the response (or dependent) variable  0 and  1 are the unknown parameters  ~ N(0,) and data: (x 1, y 1 ), (x 2, y 2 ),..., (x n, y n ) Linear Regression Model

5 Some Linearizing Transformations

6 Least squares estimates of  0 and  1

7 Point estimate of the linear model is Least squares regression equation

8 The data used for illustration are from a study of two methods of estimating tread wear of commercial tires. The data are shown here and plotted. The variable which is taken as the independent variable X is the estimated tread life in hundreds of miles by the weight-loss method. The associated variable Y is the estimated tread life by the groove-depth method. Simple Linear Regression - Example

9

10 To calculate the regression line we calculate b 0 and b 1 from the formulas on slide 6 using the following data: n=16 Simple Linear Regression - Example

11 Simple Linear Regression - Example

12 Therefore is the equation, i.e., a point estimate of the straight line Simple Linear Regression - Example

13 Point estimate of  2

14 Using the data from the previous example on slide 8, and the following formula, we calculate and Simple Linear Regression - Example

15 (1 - ) 100% confidence interval for  0 is where and where Interval Estimates for y intercept (  0 )

16 (1 - ) 100% confidence interval for  1 is where and where Interval Estimates for slope (  1 )

17 Confidence interval for conditional mean of Y, given x and

18 Using the data from the previous example on slide 8, and the formulas on the previous slide, we can calculate the confidence interval. A 95% confidence is used. Confidence interval for conditional mean of Y, given x - Example

19 Prediction interval for a single future value of Y, given x and

20 For the example, a 95% confidence prediction interval (shown in blue) is determined. Prediction interval for a single future value of Y, given x - Example

21 Prediction interval for a single future value of Y, given x - Example