Modeling a Linear RelationshipBivariate DataScatterplotsCase Study 15Scatter PlotExampleDescribing a RelationshipLinear AssociationFree-Lunch Participation vs. Graduation RateSlide 10Teachers’ Salary vs. Graduation RateSlide 12Slide 13Passing Rate on English SOL vs. Graduation RateSlide 15Slide 16Strong vs. Weak AssociationStrong Positive Linear AssociationSlide 19Weak Positive Linear AssociationSlide 21Slide 22TI-83 - ScatterplotsSlide 24Slide 25Simple Linear RegressionMeasuring the Goodness of FitSlide 28Slide 29Slide 30Slide 31Slide 32ResidualsSlide 34Slide 35Slide 36Slide 37Slide 38Slide 39Least Squares LineSlide 41Slide 42Slide 43Slide 44TI-83 – Computing ResidualsSum of Squared ResidualsSlide 47Slide 48Slide 49The Line y^ = 8 + xThe Line y^ = 7.3 + 1.1xSlide 52Slide 53PredictionInterpolation vs. ExtrapolationModeling a Linear RelationshipLecture 44Secs. 13.1 – 13.3.1Wed, Nov 28, 2007Bivariate DataData 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.ScatterplotsScatterplot – A display in which each observation (x, y) is plotted as a point in the xy plane.Case Study 15District Free Lunch Grad. Rate District Free Lunch Grad. RateAmelia 41.2 68.9 King and Queen 59.9 64.1Caroline 40.2 62.9 King William 27.9 67.0Charles City 45.8 67.7 Louisa 44.9 80.1Chesterfield 22.5 80.5 New Kent 13.9 77.0Colonial Hgts 25.7 73.0 Petersburg 61.6 54.6Cumberland 55.3 63.9 Powhatan 12.2 89.3Dinwiddie 45.2 71.4 Prince George 30.9 85.0Goochland 23.3 76.3 Richmond 74.0 46.9Hanover 13.7 90.1 Sussex 74.8 59.0Henrico 30.2 81.1 West Point 19.1 82.0Hopewell 63.1 63.4Scatter PlotFree LunchRateGraduation Rate20 30 40 50 60 70 805060809070ExampleDoes there appear to be a relationship?How can we tell?Describing a RelationshipHow would we describe the relationship between the free-lunch participation and the graduation rate?Linear AssociationDraw (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 RateFree LunchRateGraduation Rate20 30 40 50 60 70 805060809070Free-Lunch Participation vs. Graduation RateFree LunchRateGraduation Rate20 30 40 50 60 70 805060809070Teachers’ Salary vs. Graduation RateDistrict Avg Salary Grad. Rate District Avg Salary Grad. RateAmelia 30446 68.9 King and Queen 38803 64.1Caroline 41935 62.9 King William 42750 67.0Charles City 39530 67.7 Louisa 39010 80.1Chesterfield 44417 80.5 New Kent 39891 77.0Colonial Hgts 48999 73.0 Petersburg 38252 54.6Cumberland 39380 63.9 Powhatan 41523 89.3Dinwiddie 42866 71.4 Prince George 44529 85.0Goochland 41893 76.3 Richmond 45875 46.9Hanover 42715 90.1 Sussex 44142 59.0Henrico 45021 81.1 West Point 40797 82.0Hopewell 42351 63.4Teachers’ Salary vs. Graduation RateAverageTeacher’sSalary(1000’s)Graduation Rate50608090703040 50Teachers’ Salary vs. Graduation RateAverageTeacher’sSalary(1000’s)Graduation Rate50608090703040 50Passing Rate on English SOL vs. Graduation RateDistrict Eng SOL Grad. Rate District Eng SOL Grad. RateAmelia 77 68.9 King and Queen 62 64.1Caroline 73 62.9 King William 69 67.0Charles City 69 67.7 Louisa 74 80.1Chesterfield 81 80.5 New Kent 81 77.0Colonial Hgts 68 73.0 Petersburg 39 54.6Cumberland 81 63.9 Powhatan 86 89.3Dinwiddie 73 71.4 Prince George 75 85.0Goochland 88 76.3 Richmond 59 46.9Hanover 84 90.1 Sussex 51 59.0Henrico 81 81.1 West Point 96 82.0Hopewell 73 63.4Passing Rate on English SOL vs. Graduation RatePassing Rateon English SOLGraduation Rate506080907050 60 70 80 9010040Passing Rate on English SOL vs. Graduation RatePassing Rateon English SOLGraduation Rate506080907050 60 70 80 9010040Strong vs. Weak AssociationThe association is strong if the oval is narrow.The association is weak if the oval is wide.Strong Positive Linear AssociationxyStrong Positive Linear AssociationxyWeak Positive Linear AssociationxyWeak Positive Linear AssociationxyExampleDraw a scatterplot of the following data.x y1 83 124 95 148 169 2011 1715 24TI-83 - ScatterplotsTo 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 - ScatterplotsThe 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 - ScatterplotsTo 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 RegressionTo 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 FitWhich line better fits the data?xy0 5 1510102015Measuring the Goodness of FitWhich line better fits the data?xy0 5 1510102015Measuring the Goodness of FitStart with the scatterplot.xy0 5 1510102015Measuring the Goodness of FitDraw any line through the scatterplot.xy0 5 1510102015Measuring the Goodness of FitMeasure the vertical distances from every point to the line.xy0 5 1510102015Measuring the Goodness of FitEach of these distances is called a residual.xy0 5 1510102015ResidualsThe formula for the residuals iswhere y is the observed value and y^ is the value predicted by the model.yyeˆResidualsNotice 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.ResidualsThe residuals represent errors in the predicted y.xy0 5 1510102015ResidualsThe residuals represent errors in the predicted y.xy0 5 1510102015y^yxResidualsThe residuals represent errors in the predicted y.xy0 5 1510102015xerrorMeasuring the Goodness of FitThe line of best fit is the line with the smallest possible sum of squared residuals.ExampleConsider the
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