A Madness for all Seasons

Our fourth post on the  2017 VCE exam madness will be similar to our previous post: a quick whack of a straight-out error. This error was flagged by a teacher friend, David. (No, not that David.)

The 11th multiple choice question on the first Further Mathematics Exam reads as follows:

Which one of the following statistics can never be negative? 

A. the maximum value in a data set

B. the value of a Pearson correlation coefficient

C. the value of a moving mean in a smoothed time series

D. the value of a seasonal index

E. the value of a slope of a least squares line fitted to a scatterplot

Before we get started, a quick word on the question’s repeated use of the redundant “the value of”.

Bleah!

Now, on with answering the question.

It is pretty obvious that the statistics in A, B, C and E can all be negative, so presumably the intended answer is D. However, D is also wrong: a seasonal index can also be negative. Unfortunately the explanation of “seasonal index” in the standard textbook is lost in a jungle of non-explanation, so to illustrate we’ll work through a very simple example.

Suppose a company’s profits and losses over the four quarters of a year are as follows:

    \[ \begin{tabular} {| c | c | c | c |}\hline {\bf\phantom{S}Summer \phantom{I}} &{\bf\phantom{S}Autumn \phantom{I}} &{\bf\phantom{S}Winter \phantom{I}} &{\bf\phantom{S}Spring \phantom{I}} \\  \hline {\bf \$6000} & {\bf -\$1000} & {\bf -\$2000} & {\bf \$5000}\\ \hline \end{tabular}\]

So, the total profit over the year is $8,000, and then the average quarterly profit is $2000. The seasonal index (SI) for each quarter is then that quarter’s profit (or loss) divided by the average quarterly profit:

    \[ \begin{tabular} {| c | c | c | c |}\hline {\bf Summer SI} &{\bf Autumn SI} &{\bf Winter SI} &{\bf Spring SI} \\  \hline {\bf 3} & {\bf -0.5} & {\bf -1.0} & {\bf 2.5}\\ \hline \end{tabular}\]

Clearly this example is general, in the sense that in any scenario where the seasonal data are both positive and negative, some of the seasonal indices will be negative. So, the exam question is not merely technically wrong, with a contrived example raising issues: the question is wrong wrong.

Now, to be fair, this time the VCAA has a defense. It appears to be more common to apply seasonal indices in contexts where all the data are one sign, or to use absolute values to then consider magnitudes of deviations. It also appears that most or all examples Further students would have studied included only positive data.

So, yes, the VCAA (and the Australian Curriculum) don’t bother to clarify the definition or permitted contexts for seasonal indices. And yes, the definition in the standard textbook implicitly permits negative seasonal indices. And yes, by this definition the exam question is plain wrong. But, hopefully most students weren’t paying sufficient attention to realise that the VCAA weren’t paying sufficient attention, and so all is ok.

Well, the defense is something like that. The VCAA can work on the wording.

 

Further Madness

Our third post on the 2017 VCE exam madness will be brief, on a question containing a flagrant error.

The first question in the matrix module of Further Mathematics’ Exam 2 is concerned with a school canteen selling pies, rolls and sandwiches over three separate weeks. The number of items sold is set up as a 3 x 3 matrix, one row for each week and one column for each food choice. The last part, (c)(ii), of the question then reads:

The matrix equation below shows that the total value of all rolls and sandwiches sold in these three weeks is $915.60 

    \[   \boldsymbol{L \times\begin{bmatrix} 491.55 \\ 428.00\\ 487.60 \end{bmatrix} \ = \ [915.60]}\]

Matrix L in this equation is of order 1 x 3.

Write down matrix L.

This 1-mark question is presumably meant to be a gimme, with answer L = [0 1 1]. Unfortunately the question is both weird and wrong. (And lacking in punctuation. Guys, it’s not that hard.) The wrongness comes from the examiners having confused their rows and columns. As is made clear in the the previous part, (c)(i), of the question, the  3 x 1 matrix of numbers indicates the total earnings from each of the three weeks, not from each of the three food choices. So, the equation indicates the total value of all products sold in weeks 2 and 3.

There’s not much to say about such an obvious error. It is very easy to confuse rows and columns, and we’ve all done it on occasion, but if VCAA’s vetting cannot catch this kind of mistake then it cannot be relied upon to catch anything. The only question is how the Examiners’ Report will eventually address the error. The VCAA is well-practised in cowardly silence and weasel-wording, but it would be exceptionally Trumplike to attempt such tactics here.

Error aside, the question is artificial, and it is not clear that the matrix equation “shows” much of anything. Yes, 0-1 and on-or-off matrices are important and useful, but the use of such a matrix in this context is contrived and confusing. Not a hanging offence, and benign by VCAA’s standards, but the question is pretty silly. And, not forgetting, wrong.