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Two-Variable Data Analysis 103
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solution: (a) Using the calculator (LinReg(a+bx) L1, L2, Y1), we find height = 6494 + 0634(age), r = 0993 The large value of r tells us that the points are close to a line The scatterplot and LSLR are shown below on the graph at the left
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Age Scatter plot and LSRL for predicting height from age
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Residuals
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Age Scatter plot of residuals vs age
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From the graph on the left, a line appears to be a good fit for the data (the points lie close to the line) The residual plot on the right shows no readily obvious pattern, so we have good evidence that a line is a good model for the data and we can feel good about using the LSRL to predict height from age (b) The residual (actual minus predicted) for age = 19 months is 771 (6494 + 0634 19) = 0114 Note that 771 Y1(19) = 0112 (Note that you can generate a complete set of residuals, which will match what is stored in RESID, in a list Assuming your data are in L1 and L2 and that you have found the LSRL and stored it in Y1, let L3 = L2 Y1(L1) The residuals for each value will then appear in L3 You might want to let L4 = RESID (by pasting RESID from the LIST menu) and observe that L3 and L4 are the same Digression: Whenever we have graphed a residual plot in this section, the vertical axis has been the residuals and the horizontal axis has been the x-variable On some computer printouts, you may see the horizontal axis labeled Fits (as in the graph below) or Predicted Value
30 x x x x 0 x x x x x
RESI1
x 30 30 45 60 x 75 90
FITS2
What you are interested in is the visual image given by the residual plot, and it doesn t matter if the residuals are plotted against the x-variable or something else, like FITS2 the scatter of the points above and below 0 stays the same All that changes are the horizontal distances between points This is the way it must be done in multiple regression, since there is more than one independent variable and, as you can see, it can be done in simple linear regression
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If we are trying to predict a value of y from a value of x, it is called interpolation if we are predicting from an x-value within the range of x-values It is called extrapolation if we are predicting from a value of x outside of the x-values example: Using the age/height data from the previous example, we are interpolating Age 18 19 20 21 22 23 24 25 813 26 27 28 29
Height 76
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if we attempt to predict height from an age between 18 and 29 months It is interpolation if we try to predict the height of a 205-month-old baby We are extrapolating if we try to predict the height of a child less than 18 months old or more than 29 months old If a line has been shown to be a good model for the data and if it fits the line well (ie, we have a strong r and a more or less random distribution of residuals), we can have confidence in interpolated predictions We can rarely have confidence in extrapolated values In the example above, we might be willing to go slightly beyond the ages given because of the high correlation and the good linear model, but it s good practice not to extrapolate beyond the data given If we were to extrapolate the data in the example to a child of 12 years of age (144 months), we would predict the child to be 1562 inches, or more than 13 feet tall!
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