Chapter 1.5, Problem 48E

Solution

To state: Why is it important for fairness that the scaling function used to scale the raw marks to grades be an increasing function.

To determine: What would the scaling function in exercise $47$ do for a student who does enough “extra credit” problems to get a raw score of $136$ .

It is important for fairness that the scaling function used to scale the raw marks to grades be an increasing function because otherwise the students with lower raw marks will be graded above the students with higher raw marks.

The student with raw marks $136$ obtains a grade of $133$ . Thus, even though the scaling function is increasing it is not necessary that every score will go up.

Known from the previous question $47$:

The linear equation $y=31+\frac{3}{4}x$ can be used to convert raw scores to scaled grades. Here, $y$ represents the scaled grades and $x$ represents the raw scores

Concepts Used:

For fairness the students with larger raw scores must be awarded a larger scaled grade.

Substitution of a variable in an expression.

Calculations:

If a student is awarded $136$ raw marks then the corresponding grade can be obtained by substituting $x=136$ in the equation $y=31+\frac{3}{4}x$ and evaluating $y$ .

  $\begin{array}{l}y=31+\frac{3}{4}x\\ y=31+\frac{3}{4}\left(136\right)\\ y=31+3\left(34\right)\\ y=31+102\\ y=133\end{array}$

Conclusion:

It is important for fairness that the scaling function used to scale the raw marks to grades be an increasing function because otherwise the students with lower raw marks will be graded above the students with higher raw marks.

The student with raw marks $136$ obtains a grade of $133$ . Thus, even though the scaling function is increasing it is not necessary that every score will go up.