1. Simple Linear Regression Statistics 700 Week of November 27

2. Week of 11/27/2000Simple Linear Regression2 Example for Illustration The human body takes in more oxygen when exercising than when it is at rest. To deliver oxygen to the muscles, the heart must beat faster. Heart rate is easy to measure, but measuring oxygen uptake requires elaborate equipment. If oxygen uptake (VO 2 ) can be accurately predicted from heart rate (HR), the predicted values may replace actually measured values for various research purposes. Unfortunately, not all human bodies are the same, so no single prediction equation works for all people. Researchers can, however, measure both HR and VO 2 for one person under varying sets of exercise conditions and calculate a regression equation for predicting that person’s oxygen uptake from heart rate.

3. Week of 11/27/2000Simple Linear Regression3 Data From An Individual Goals in this illustration: Scatterplot: linear relationship or not? Obtain the best-fitting line using least-squares. To test whether the model is significant or not. To obtain a confidence interval for the regression coefficient. To obtain predictions.

4. Week of 11/27/2000Simple Linear Regression4 The Scatterplot

5. Week of 11/27/2000Simple Linear Regression5 Simple Linear Regression Model 1. Conditional on X=x, the response variable Y has mean equal to  x  x  2.  is the y-intercept; while  is the slope of the regression line, which could be interpreted as the change in the mean value per unit change in the independent variable. 3. For each X = x, the conditional distribution of Y is normal with mean  (x) and variance  2. 4. Y 1, Y 2, …, Y n are independent of each other. Shorthand: Y i =  +  x i +  i with  i IID N(0,  2 )

6. Week of 11/27/2000Simple Linear Regression6

7. Week of 11/27/2000Simple Linear Regression7

8. Week of 11/27/2000Simple Linear Regression8

9. Week of 11/27/2000Simple Linear Regression9

10. Week of 11/27/2000Simple Linear Regression10 Results of Regression Analysis (using Minitab)

11. Week of 11/27/2000Simple Linear Regression11 Fitted Line on the Scatterplot

12. Week of 11/27/2000Simple Linear Regression12