We will look for the value of m that minimizes sae b

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We will look for the value of m that minimizes SAE. (b) Open the Excel spreadsheet document Heights.xls . Notice that the 20 heights are in column A (sorted), and column B contains candidates for m from 65 to 70, in increments of 0.01. Click on cell C2, and notice that it contains a formula for SAE( m ), which refers to the data values in column A and to the value of m from B2 (namely, m = 65 in that cell). Use Excel’s “fill down” featur e to calculate the values of SAE( m ) for the remainder of column C. [ Hint : You can do this in one of several ways: With the C2 cell selected, double click on the box in the lower right corner, or pull the right corner of the highlighted C2 cell down to the end of the column, or highlight C2 and the cells to be filled and choose Edit> Fill> Down.] Confirm that the SAE value for the sample mean (67.75) agrees with what you calculated in (a).
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Chance/Rossman, 2015 ISCAM III Exploration 376 (c) To produce a graph of SAE as a function of m : Highlight columns B and C. Choose Insert > Scatter and then choose the first option in the second row (Scatter with Smooth Lines) to construct a graph of this function. [ Hint : After you create the graph, you may want to change the x -min and x -max values by double clicking on the horizontal axis and changing the Minimum and Maximum under Axis Options.] Reproduce a rough sketch of this graph below, and comment on the shape of this function. Does it seem to follow a familiar form, such as linear or parabolic or exponential? Explain. (d) Does there seem to be a unique value of m that minimizes this SAE( m ) function? If so, identify it. If not, describe all values of m that minimize the function. Also report the value of the SAE for this optimal value of m . [ Hint : Search through the entries in column C to find the smallest value, and then read off the corresponding value in column B.] Optimal m = SAE at the optimal m = (e) Look through the sorted heights in column A. What do you notice about where this optimal value of m is located in the list of sorted heights? Does this remind you of a familiar statistic? (f) Suppose the tenth largest height (in row 11) had been 66 inches instead of 67. Make this change to the spreadsheet (cell A11), and note that the graph of the SAE function is updated automatically. Now is there a unique value of m that minimizes this SAE( m ) function? If so, identify it. If not, describe all values of m that minimize the function, and also report the (optimal, minimum) value of the SAE. Optimal m = SAE at the optimal m = (g) Return the tenth largest height back to 67 inches. Now suppose the tallest person in the sample had been 80 inches tall instead of 77. Make this change to the spreadsheet (cellA21) and note how the graph of the SAE function changes. Also report the value of the SAE. Optimal m = SAE at the optimal m =
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Chance/Rossman, 2015 ISCAM III Exploration 377 (h) Based on these findings, make a conjecture as to how one can determine, from a generic set of 20 data values, the value that minimizes the sum of the absolute prediction errors from the data values.
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