Modeling a Linear Relationship Lecture 44 Secs 13 1 13 3 1 Wed Nov 28 2007 Bivariate Data Data is called bivariate if each observations consists of a pair of values x y x is the explanatory variable y is the response variable x is also called the independent variable y is also called the dependent variable Scatterplots Scatterplot A display in which each observation x y is plotted as a point in the xy plane Case Study 15 District Free Lunch Grad Rate Amelia 41 2 68 9 Caroline 40 2 Charles City District Free Lunch Grad Rate King and Queen 59 9 64 1 62 9 King William 27 9 67 0 45 8 67 7 Louisa 44 9 80 1 Chesterfield 22 5 80 5 New Kent 13 9 77 0 Colonial Hgts 25 7 73 0 Petersburg 61 6 54 6 Cumberland 55 3 63 9 Powhatan 12 2 89 3 Dinwiddie 45 2 71 4 Prince George 30 9 85 0 Goochland 23 3 76 3 Richmond 74 0 46 9 Hanover 13 7 90 1 Sussex 74 8 59 0 Henrico 30 2 81 1 West Point 19 1 82 0 Hopewell 63 1 63 4 Scatter Plot Graduation Rate 90 80 70 60 50 20 30 40 50 60 70 80 Free Lunch Rate Example Does there appear to be a relationship How can we tell Describing a Relationship How would we describe the relationship between the free lunch participation and the graduation rate Linear Association Draw or imagine an oval around the data set If the oval is tilted then there is some linear association If the oval is tilted upwards from left to right then there is positive association If the oval is tilted downwards from left to right then there is negative association If the oval is not tilted at all then there is no association Free Lunch Participation vs Graduation Rate Graduation Rate 90 80 70 60 50 20 30 40 50 60 70 80 Free Lunch Rate Free Lunch Participation vs Graduation Rate Graduation Rate 90 80 70 60 50 20 30 40 50 60 70 80 Free Lunch Rate Teachers Salary vs Graduation Rate District Avg Salary Grad Rate Amelia 30446 68 9 Caroline 41935 Charles City District Avg Salary Grad Rate King and Queen 38803 64 1 62 9 King William 42750 67 0 39530 67 7 Louisa 39010 80 1 Chesterfield 44417 80 5 New Kent 39891 77 0 Colonial Hgts 48999 73 0 Petersburg 38252 54 6 Cumberland 39380 63 9 Powhatan 41523 89 3 Dinwiddie 42866 71 4 Prince George 44529 85 0 Goochland 41893 76 3 Richmond 45875 46 9 Hanover 42715 90 1 Sussex 44142 59 0 Henrico 45021 81 1 West Point 40797 82 0 Hopewell 42351 63 4 Teachers Salary vs Graduation Rate Graduation Rate 90 80 70 60 50 30 40 50 Average Teacher s Salary 1000 s Teachers Salary vs Graduation Rate Graduation Rate 90 80 70 60 50 30 40 50 Average Teacher s Salary 1000 s Passing Rate on English SOL vs Graduation Rate District Eng SOL Grad Rate Amelia 77 68 9 Caroline 73 Charles City District Eng SOL Grad Rate King and Queen 62 64 1 62 9 King William 69 67 0 69 67 7 Louisa 74 80 1 Chesterfield 81 80 5 New Kent 81 77 0 Colonial Hgts 68 73 0 Petersburg 39 54 6 Cumberland 81 63 9 Powhatan 86 89 3 Dinwiddie 73 71 4 Prince George 75 85 0 Goochland 88 76 3 Richmond 59 46 9 Hanover 84 90 1 Sussex 51 59 0 Henrico 81 81 1 West Point 96 82 0 Hopewell 73 63 4 Passing Rate on English SOL vs Graduation Rate Graduation Rate 90 80 70 60 50 40 50 60 70 80 90 100 Passing Rate on English SOL Passing Rate on English SOL vs Graduation Rate Graduation Rate 90 80 70 60 50 40 50 60 70 80 90 100 Passing Rate on English SOL Strong vs Weak Association The association is strong if the oval is narrow The association is weak if the oval is wide Strong Positive Linear Association y x Strong Positive Linear Association y x Weak Positive Linear Association y x Weak Positive Linear Association y x Example Draw a scatterplot of the following data x y 1 8 3 12 4 9 5 14 8 16 9 20 11 17 15 24 TI 83 Scatterplots To set up a scatterplot Enter the x values in L1 Enter the y values in L2 Press 2nd STAT PLOT Select Plot1 and press ENTER TI 83 Scatterplots The Stat Plot display appears Select On and press ENTER Under Type select the first icon a small image of a scatterplot and press ENTER For XList enter L1 For YList enter L2 For Mark select the one you want and press ENTER TI 83 Scatterplots To draw the scatterplot Press ZOOM The Zoom menu appears Select ZoomStat 9 and press ENTER The scatterplot appears Press TRACE and use the arrow keys to inspect the individual points Simple Linear Regression To quantify the linear relationship between x and y we wish to find the equation of the line that best fits the data Typically there will be many lines that all look pretty good How do we measure how well a line fits the data Measuring the Goodness of Fit Which line better fits the data y 20 15 10 x 0 5 10 15 Measuring the Goodness of Fit Which line better fits the data y 20 15 10 x 0 5 10 15 Measuring the Goodness of Fit Start with the scatterplot y 20 15 10 x 0 5 10 15 Measuring the Goodness of Fit Draw any line through the scatterplot y 20 15 10 x 0 5 10 15 Measuring the Goodness of Fit Measure the vertical distances from every pointy to the line 20 15 10 x 0 5 10 15 Measuring the Goodness of Fit Each of these distances is called a residual y 20 15 10 x 0 5 10 15 Residuals The formula for the residuals is e y y where y is the observed value and y is the value predicted by the model Residuals Notice that the residual is positive if the data point is above the line and it is negative if the data point is below the line Residuals The residuals represent errors in the predicted y y 20 15 10 x 0 5 10 15 Residuals The residuals represent errors in the predicted y y y 20 y 15 10 x 0 5 10 x 15 Residuals The residuals represent errors in the predicted y y error 20 15 10 x 0 5 10 x 15 Measuring the Goodness of Fit The line of best fit is the line with the smallest possible sum of squared residuals Example Consider the data points x y 1 8 3 12 4 9 5 14 8 16 9 20 11 17 15 24 Least Squares Line Let s see how good the fit is for the line y 8 x y …
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