1. Create a scatterplot of this data. 2. Find the correlation coefficient.

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2 How Fast Foods Compare Company Entree Total Calories Fat (grams) McDonald s Big Mac Filet o Fish Burger King Whopper Big Fish Sandwich Wendy s Single Burger Create a scatterplot of this data. 2. Find the correlation coefficient. Premium Filet of Fish Arby s Regular Roast Beef French Dip Sub Taco Bell Soft Taco Nacho BellGrande Write a few sentences about what the plot says about how these fast foods compare. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 7-2

3 Chapter 8 Linear Regression Copyright 2010, 2007, 2004 Pearson Education, Inc.

4 Fat Versus Protein: An Example The following is a scatterplot of total fat versus protein for 30 items on the Burger King menu: Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-4

5 The Linear Model The correlation in this example is It says The linear association between these two variables is fairly strong, but it doesn t tell what the line is. We can say more about the linear relationship between two quantitative variables with a model. A model simplifies reality to help us understand underlying patterns and relationships (like the normal model) and gives us an equation to work with. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-5

6 The Linear Model (cont.) The linear model is an equation of a straight line through the data. The points in the scatterplot don t all line up, but a straight line can summarize the general pattern. The linear model can help us understand how the values are associated and help us make predictions. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-6

7 Residuals The model won t be perfect, regardless of the line we draw. Some points will be above the line and some will be below. The estimate made from a model is the predicted value denoted as (y-hat). ŷ Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-7

8 Residuals (cont.) The difference between the observed value and its associated predicted value is called the residual. To find the residuals, we always subtract the predicted value from the observed one: residual observed predicted y ŷ Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-8

9 Residuals (cont.) A negative residual means the predicted value s too big (an overestimate). A positive residual means the predicted value s too small (an underestimate). In the figure, the estimated fat of the BK Broiler chicken sandwich is 36 g, while the true value of fat is 25 g, so the residual is = 11 g of fat. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-9

10 Best Fit Means Least Squares Some residuals are positive, others are negative, and, on average, they cancel each other out. So, we can t assess how well the line fits by adding up all the residuals. Similar to what we did with deviations, we square the residuals and add the squares. The smaller the sum, the better the fit. The line of best fit is the line for which the sum of the squared residuals is smallest, the least squares line. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-10

11 Correlation and the Line The figure shows the scatterplot of z-scores for fat and protein. If a burger has average protein content, it should have about average fat content too. The point ( x, y) is on the line. Moving one standard deviation away from the mean in x moves us r standard deviations away from the mean in y. r is the correlation coefficient Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-11

12 Correlation and the Line (cont.) Put generally, moving any number of standard deviations away from the mean in x, moves us r times that number of standard deviations away from the mean in y. Usually we think of the equation of a line as y = mx + b. Now we can think of our equation as z y = mz x because we are using standardized points, not (x, y) and it turns our the best slope (m) of our line is r. So we have the equation z y = rz x So our equation for the fat vs. protein example would be z fat = 0.83z protein since r = 0.83 Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-12

13 Example (p. 174 Just Checking) A scatterplot of house Size (in thousands of square feet) vs. house Price (in thousands of dollars) for houses sold recently in Saratoga, NY shows a relationship that is straight, with only moderate scatter and no outliers. The correlation between house Price and Size is a) You go to an open house and find that the house is 1 standard deviation above the mean in size. What would you guess about its price? Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-13

14 Example (continued) A scatterplot of house Price (in thousands of dollars) vs. house Size (in thousands of square feet) for houses sold recently in Saratoga, NY shows a relationship that is straight, with only moderate scatter and no outliers. The correlation between house Price and Size is b) You read an ad for a house priced 2 standard deviations below the mean. What would you guess about its size? Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-14

15 Example (continued) A scatterplot of house Price (in thousands of dollars) vs. house Size (in thousands of square feet) for houses sold recently in Saratoga, NY shows a relationship that is straight, with only moderate scatter and no outliers. The correlation between house Price and Size is c) A friend tells you about a house whose size in square meters (he s European) is 1.5 standard deviations above the mean. What would you guess about its size in square feet? Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-15

16 How Big Can Predicted Values Get? As previous stated: moving any number of standard deviations away from the mean in x, moves us r times that number of standard deviations away from the mean in y r cannot be bigger than 1 (in absolute value), so each predicted y tends to be closer to its mean (in standard deviations) than its corresponding x was. If you move 1 SD away from the mean in x you will most likely move less than 1 SD away from the mean in y. This property of the linear model is called regression to the mean; the line is called the regression line. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-16

17 Homework Read p Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-17

18 The Regression Line in Real Units The regression line models the equation in the original units whereas the least squares line is in standardized units. Remember from Algebra that a straight line can be written as: In Statistics we use a slightly different notation: ŷ y mx b ŷ b 0 b 1 x We write to emphasize that the points that satisfy this equation are just our predicted values, not the actual data values. This model says that our predictions from our model follow a straight line. If the model is a good one, the data values will scatter closely around it. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-18

19 The Regression Line in Real Units(cont.) We write b 1 and b 0 for the slope and intercept of the line. b 1 is the slope, which tells us how ŷ changes with respect to x. b 0 is the y-intercept, which tells where the line crosses (intercepts) the y-axis. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-19

20 The Regression Line in Real Units (cont.) In our model, we have a slope (b 1 ): The slope is built from the correlation and the standard deviations: b 1 r s y s x up r SDs in y over 1 SD in x Our slope is always in units of y per unit of x. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-20

21 Fat Versus Protein: An Example We know that our model predicts that for each increase of one SD in protein we ll see an increase of about 0.83 SD in fat since r = To find the slope and y-intercept we need to find the mean and SD of protein and fat. ( x, s x, y, s y ) To model y in terms of x (predict fat from protein), the slope is b 1 = rs y s x b 1 = rs y s x = g fat (14.0) g protein Protein: x = 17.2 g s x = 14.0 g Fat: y = 23.5 g s y = 16.4 g = 0. 97g of fat per g of protein Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-21

22 The Regression Line in Real Units (cont.) In our model, we also have an intercept (b 0 ). The intercept is built from the means and the slope: ŷ b 0 b 1 x b 0 y b 1 x Our intercept is always in units of y. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-22

23 Fat Versus Protein: An Example We need to find the y-intercept, b 0. The intercept is built from our means and the slope. b 0 y b 1 x b 0 = 23.5g fat 0.97 g fat g protein 17.2 g protein = 6.8 g fat Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-23

24 Fat Versus Protein: An Example Putting the slope and y-intercept into the equation we get: This means that every additional gram of protein is associated with an additional 0.97 grams of fat. Or - BK sandwiches have about 0.97 g of fat per g of protein. The predicted fat content for a BK Broiler chicken sandwich (with 30 g of protein) is (30) = 35.9 grams of fat. We are back to our original units! Yay! Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-24

25 Slope: We know that changing the units of the variables doesn t change the correlation. But slope is in terms of y and x (we re back to our original units, not standardized values). Changing the units changes the SD. You need to be mindful of how you choose to express x and y what units to use. Do you want kilobytes per second or seconds per kilobyte? Units of slope are always the units of y per unit of x. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-25

26 Interpreting b 0 - y-intercept The line of regression for the BK sandwiches example is: How would we interpret the y-intercept of 6.8g of fat? A BK sandwich with g of protein would have, on average, about g of fat. Is this reasonable? The intercept serves as a starting value for our predictions but we don t interpret it as a meaningful predicted value. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-26

27 The Regression Line in Real Units (cont.) Since regression and correlation are closely related, we need to check the same conditions for regressions as we did for correlations: Quantitative Variables Condition Straight Enough Condition Outlier Condition Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-27

28 Residuals Revisited The linear model assumes that the relationship between the two variables is a perfect straight line. The residuals are the part of the data that hasn t been modeled. or (equivalently) Or, in symbols, Data = Model + Residual Residual = Data Model e y ŷ Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-28

29 Residuals Revisited (cont.) Residuals help us to see whether the model makes sense. When a regression model is appropriate, nothing interesting should be left behind. After we fit a regression model, we usually plot the residuals in the hope of finding nothing. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-29

30 Residuals Revisited (cont.) The residuals for the BK menu regression look appropriately boring: Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-30

31 Example 1 Find the line of regression for this data if r = 0.873, sx = 7.53, sy = 0.33, x = 80.13, y = 1.67 ŷ b 0 b 1 x Copyright 2010, 2007, 2004 Pearson Education, Inc. b 1 r s y Statistics School % Attendance Mean Grade Point Average Bay View Custer Hamilton Juneau King Madison Marshall School of Arts Milwaukee Tech North Division Pulaksi Riverside South Division Vincent Washington s x Slide 8-31

32 Example (p. 178 Just Checking) Let s look again at the relationship between house Price (in thousands of dollars) and house size (in thousands of square feet) in Saratoga. The regression model is: Price Size a) What does the slope of mean? b) What are the units of the slope? c) Your house is 2000 sq ft bigger than your neighbor s house. How much more do you expect it to be worth? d) Is the y-intercept of meaningful? Explain. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-32

33 Example (p. 181 Just Checking) Our linear model for Saratoga homes uses the Size (in thousands of square feet) to estimate the Price (in thousands of dollars): Price Size Suppose you re thinking of buying a home there. a) Would you prefer to find a home with a negative or positive residual? Explain Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-33

34 Example (cont) Our linear model for Saratoga homes uses the Size (in thousands of square feet) to estimate the Price (in thousands of dollars): Price Size Suppose you re thinking of buying a home there. b) You plan to look for a home of about 3000 square feet. How much should you expect to have to pay? Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-34

35 Example (cont) Our linear model for Saratoga homes uses the Size (in thousands of square feet) to estimate the Price (in thousands of dollars): Price Size Suppose you re thinking of buying a home there. c) You find a nice home that is 3000 square feet selling for $300,000. What s the residual? Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-35

36 Example 3 Find the line of best fit for this data. Time (sec) Heig ht (m) Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-37

37 Homework Read p. 181-end Chapter 8 Homework: p. 192 #1, 3, 5, 21, 37, 39 Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-38

38 The Residual Standard Deviation The standard deviation of the residuals, s e, measures how much the points spread around the regression line. Check the Equal Variance Assumption with the Does the Plot Thicken? Condition. Check to make sure the residual plot has about the same amount of scatter throughout Check the scatterplot to make sure the spread is about the same all along the line. We estimate the SD of the residuals using: s e e 2 n 2 Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-39

39 The Residual Standard Deviation We don t need to subtract the mean because the mean of the residuals e 0. If you make a histogram or normal probability plot of the residuals. It should look unimodal and roughly symmetric. Then we can apply the Rule to see how well the regression model describes the data. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-40

40 R 2 The Variation Accounted For The variation in the residuals is the key to assessing how well the model fits. In the BK menu items example, total fat has a standard deviation of 16.4 grams. The standard deviation of the residuals is 9.2 grams. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-41

41 R 2 The Variation Accounted For (cont.) If the correlation were 1.0 and the model predicted the fat values perfectly, the residuals would all be zero and have no variation. As it is, the correlation is 0.83 not perfection. However, we did see that the model residuals had less variation than total fat alone. We can determine how much of the variation is accounted for by the model and how much is left in the residuals. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-42

42 R 2 The Variation Accounted For (cont.) The squared correlation, r 2, gives the fraction of the data s variance accounted for by the model. Thus, 1 r 2 is the fraction of the original variance left in the residuals. For the BK model, r 2 = = 0.69, so 69% of the variability in total fat is accounted for by the model and = 0.31, so 31% of the variability in total fat has been left in the residuals. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-43

43 R 2 The Variation Accounted For (cont.) All regression analyses include this statistic, although by tradition, it is written R 2 (pronounced R-squared ). An R 2 of 0 means that none of the variance in the data is in the model; all of it is still in the residuals. When interpreting a regression model you need to Tell what R 2 means. In the BK example, 69% of the variation in total fat is accounted for by variation in the protein content. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-44

44 How Big Should R 2 Be? R 2 is always between 0% and 100%. What makes a good R 2 value depends on the kind of data you are analyzing and on what you want to do with it. The standard deviation of the residuals can give us more information about the usefulness of the regression by telling us how much scatter there is around the line. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-45

45 How Big Should R 2 Be? No universal standard 100%-80%: Great Regression 80%-50%: Good Regression 50-30%: Useful Regression 30%-0%: Poor Regression Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-46

46 Reporting R 2 Along with the slope and intercept for a regression, you should always report R 2 so that readers can judge for themselves how successful the regression is at fitting the data. Statistics is about variation, and R 2 measures the success of the regression model in terms of the fraction of the variation of y accounted for by the regression. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-47

47 Assumptions and Conditions Quantitative Variables Condition: Regression can only be done on two quantitative variables (and not two categorical variables), so make sure to check this condition. Straight Enough Condition: The linear model assumes that the relationship between the variables is linear. A scatterplot will let you check that the assumption is reasonable. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-48

48 Assumptions and Conditions (cont.) If the scatterplot is not straight enough, stop here. You can t use a linear model for any two variables, even if they are related. They must have a linear association or the model won t mean a thing. Some nonlinear relationships can be saved by reexpressing the data to make the scatterplot more linear. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-49

49 Assumptions and Conditions (cont.) It s a good idea to check linearity again after computing the regression when we can examine the residuals. Does the Plot Thicken? Condition: Look at the residual plot -- for the standard deviation of the residuals to summarize the scatter, the residuals should share the same spread. Check for changing spread in the residual scatterplot. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-50

50 Assumptions and Conditions (cont.) Outlier Condition: Watch out for outliers. Outlying points can dramatically change a regression model. Outliers can even change the sign of the slope, misleading us about the underlying relationship between the variables. If the data seem to clump or cluster in the scatterplot, that could be a sign of trouble worth looking into further. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-51

51 Reality Check: Is the Regression Reasonable? Statistics don t come out of nowhere. They are based on data. The results of a statistical analysis should reinforce your common sense, not fly in its face. If the results are surprising, then either you ve learned something new about the world or your analysis is wrong. When you perform a regression, think about the coefficients and ask yourself whether they make sense. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-52

52 Example (p. 183 Just Checking) Back to our regression of house Price (in thousands of dollars) on house Size (in thousands of square feet). The R 2 value is reported as 59.5%, and the standard deviation of the residuals is a) What does the R 2 value mean about the relationship of Price and Size? b) Is the correlation of Price and Size positive or negative? How do you know? Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-53

53 Example (cont) Back to our regression of house Price (in thousands of dollars) on house Size (in thousands of square feet). The R 2 value is reported as 59.5%, and the standard deviation of the residuals is d) You find that your house in Saratoga is worth $100,000 more than the regression model predicts. Should you be very surprised (as well as pleased)? Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-55

54 What Can Go Wrong? Don t fit a straight line to a nonlinear relationship. Beware extraordinary points (y-values that stand off from the linear pattern or extreme x-values). Don t extrapolate beyond the data the linear model may no longer hold outside of the range of the data. Don t infer that x causes y just because there is a good linear model for their relationship association is not causation. Don t choose a model based on R 2 alone. Remember a single outlier can change r and R 2 Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-56

55 What have we learned? When the relationship between two quantitative variables is fairly straight, a linear model can help summarize that relationship. The regression line doesn t pass through all the points, but it is the best compromise in the sense that it has the smallest sum of squared residuals. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-57

56 What have we learned? (cont.) The correlation tells us several things about the regression: The slope of the line is based on the correlation, adjusted for the units of x and y. For each SD in x that we are away from the x mean, we expect to be r SDs in y away from the y mean. Since r is always between 1 and +1, each predicted y is fewer SDs away from its mean than the corresponding x was (regression to the mean). R 2 gives us the fraction of the response accounted for by the regression model. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-58

57 What have we learned? The residuals also reveal how well the model works. If a plot of the residuals against predicted values shows a pattern, we should re-examine the data to see why. The standard deviation of the residuals quantifies the amount of scatter around the line. (tells us how good our model is at fitting our data) Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-59

58 What have we learned? (cont.) The linear model makes no sense unless the Linear Relationship Assumption is satisfied. Also, we need to check the Straight Enough Condition and Outlier Condition with a scatterplot. For the standard deviation of the residuals, we must make the Equal Variance Assumption. We check it by looking at both the original scatterplot and the residual plot for Does the Plot Thicken? Condition. Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-60

59 Homework Chapter 8 Homework: p. 192 # 15, 17, 19, 27, 45, 47 Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-61

60 LBF WKST #1 together HW: LBF WKST 2 LBF WKST #2 together HW: AP Stats Classwork WKST p. 247 #12,15,29,30 Ch 7/8 Quiz Copyright 2010, 2007, 2004 Pearson Education, Inc. Slide 8-62

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