I consider a new minimization criterion mean absolute

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(i) Consider a new minimization criterion: mean absolute error (MAE) criterion, which divides the SAE value by the number of observations. How would the MAE( m ) function compare to the SAE( m ) function? Would its minimum be achieved at the same value(s) of m ? Explain. Discussion : You should have found that the SAE function is piecewise linear . In other words, the function itself is not linear, but its component pieces are linear. In this case the function changes its slope precisely at the data values. The SAE function (and MAE function) are minimized at the median of the data and also at any value between (and including) the two middle values in the dataset (the 10 th and 11 th values in this case with n = 20 values). It is a convenient convention to define the median as the average (midpoint) of those two middle values. The MAE has a bit more natural interpretation the average deviation of the values in the data set from the value of m . Another criterion is to find the value of m that minimizes the sum of the squared deviations . (j) Recall from (j) of Investigation 5.8 the value of m that minimizes the sum of the squared prediction errors. Report this value of m for the height data. (k) Make sure that the data values in column A are back to their original values. Then click on cell D2 in the Excel spreadsheet, and notice that it gives the formula for the SSE (sum of squared errors) function. Fill this formula down the column and create a second graph to display the behavior of this function. [ Hints : Click on column B and then hold the mouse down when you click on column D to height just those two columns before you insert the graph. You will probably want to double click on the y -axis scale to change its minimum value.] Does this function have a recognizable form, such as a polynomial? What is its shape? At what value of m is the function minimized? Does this value look familiar? (l) Suppose there had been a clerical error in entering the height of the tallest student. Change the height of the tallest student from 77 inches to 770 and investigate the effect on the SSE function. Has the value of m that minimizes SSE changed? By how much? How does this compare to the effect of this clerical error on the SAE function and where it is minimized?
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Chance/Rossman, 2015 ISCAM III Exploration 378 Discussion: Although the SAE criteria would be more resistant to outliers, it does not always lead to a unique minimum (e.g., an even number of data values where the values of the middle pair are not equal). This has led to the more common criterion of least squares . See the exercises for investigating additional minimization criteria, such as the median of absolute errors and maximum of absolute errors. Investigation 5.9: Money-Making Movies Is there a tendency for movies that garner better reviews to also earn more money at the box office?
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