3.2 Least Squares Regression Part I: Interpreting a Regression Line & Prediction INTERPRET the slope and y intercept of a least-squares regression line. USE the least-squares regression line to predict y for a given x. CALCULATE and INTERPRET residuals and their standard deviation. EXPLAIN the concept of least squares. DETERMINE the equation of a least-squares regression line using a variety of methods. CONSTRUCT and INTERPRET residual plots to assess whether a linear model is appropriate. ASSESS how well the least-squares regression line models the relationship between two variables. DESCRIBE how the slope, y intercept, standard deviation of the residuals, and r 2 are influenced by outliers. Plot the data Look for overall pattern (DOFS) Calculate numerical summaries (r) When data falls into a regular pattern seek for a simplified model Mathematical relationship between two quantitative variables How does a response variable y change with respect to x changes? Can you make predictions of y for a given x? Is it a good regression model? Describe the graph: Don’t you hate it when you open a can of soda and some of the contents spray out of the can? Two AP®Statistics students, Kerry and Danielle, wanted to investigate if tapping on a can of soda would reduce the amount of soda expelled after the can has been shaken. For their experiment, they vigorously shook 40 cans of soda and randomly assigned each can to be tapped for 0 seconds, 4 seconds, 8 seconds, or 12 seconds. Then, after opening the can and cleaning up the mess, the students measured the amount of soda left in each can (in ml). Here are the data and a scatterplot. The scatterplot shows a fairly strong, positive linear association between the amount of tapping time and the amount remaining in the can. The line on the plot is a regression line for predicting the amount remaining from the amount of tapping time. Amount of soda remaining (ml) 0s 4s 8s 12 s 245 260 267 275 255 250 271 280 250 250 268 275 250 250 270 280 250 260 276 285 245 265 255 290 248 267 270 284 250 260 270 278 251 261 275 279 249 259 275 280 Suppose: ◦ y -> response variable (on vertical axis) ◦ x -> explanatory variable (on the horizontal axis). A regression line equation: ŷ = a + bx • ŷ (read “y hat”) ≡ predicted value of the response variable y for a given value of the explanatory variable x. • b ≡ slope, the amount by which y is predicted to change when x increases by one unit. • a ≡ y intercept, the predicted value of y when x = 0. Regression Line equation: price = 38257- 0.1629(miles driven) 45000 PROBLEM: Identify the slope and y intercept of the regression line. 40000 Price (in dollars) 35000 30000 25000 Interpret each value in context. 20000 15000 10000 5000 0 20000 40000 60000 80000 100000 Miles driven 120000 140000 160000 The equation of the regression line in the previous Alternate Example is soda = 248.6 +2.63 (tapping time) Problem: Identify the slope and y intercept of the regression line. Interpret each value in context. Use the regression line to predict price for a Ford F150 with 100,000 miles driven. price = 38257- 0.1629(miles driven) What are Regression Lines used for? Accuracy? Can you predict a response ŷ for any of the explanatory variable x? What is considered a Good Regression Line? A good regression line makes the vertical distances of the points from the line as small as possible. residual residual = observed y – predicted y residual = y – ŷ If the residual is +ve/-ve? Common Errors: 1. Not stating that the slope is the predicted (estimated/expected value) change in the y variable for each increase/decrease of 1 unit in the x variable. Is it accurate to say the following? Explain: “The price will go down by 0.1629 dollars for each additional mile driven” 1. 2. Check your Understanding Page 168 Activity: page 170 Investigating Properties of the Least Square Regression Line LiST THEM on the whiteboard 3. Page 171: Least Squares Regression Lines on the Calculator Brainstorm with your group and provide a quick summary on the white board Have examples to ensure your understanding on the topic Homework: page 193 # 35-42 ALL
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