A Calculated Approach

The Age

Monday September 8, 2008

Rob Vermay - Rob Vermay is a VCE assessor

Be familiar with your calculator or CAS before you enter the exam room, writes Rob Vermay

EXAMINERS assume students have access to an approved graphics calculator or CAS. This technology is useful for the modules as well as the data analysis core.

In 2007, examination questions for the data analysis core that could be readily completed with the functions on a graphics calculator or CAS included Question 9 in Examination 1 and Question 3b in Examination 2.

In Question 9, a log (y) transformation had to be applied to nine data points and the least squares regression line fitted to the transformed data.

These nine points from the given table had to be entered into the list facility of the calculator as shown in Figure 1 applicable to a TI-84 Plus calculator. Other brands have models with similar functionality. As x is the independent variable, this is entered into L1 and the dependent variable, y, is entered into L2.

Figure 1 - Given data entered into L1 and L2.

A log (y) transformation must then be applied to each of the values in L2.

The easiest way to do this is to define the whole column of L3 as log (L2).

Place the cursor to highlight the heading of the column for L3.

Now enter the expression shown in the bottom row in Figure 1. Press the LOG button and then L2 inside brackets as also shown in Figure 1.

You will find L2 above the button for the number 2 on the calculator.

Press ENTER to evaluate all log (y) values simultaneously as shown in Figure 2.

Figure 2 - Calculation of all log (y) values You must use the columns L1 and L3 for the least squares regression line calculation. Use the STAT-CALC function to enter LinReg(a+bx) L1, L3 Notice that the dependent variable is now log (y) in L3 rather than in just plain y in L2.

This produces the screen shown in Figure 3.

Figure 3 - Least squares regression equation for Question 9 You need to interpret this as log (y) = 0.88 + 0.23 x. This is alternative D in the question.

In Examination 2, Question 3b also asked for the equation to a least squares regression line for a set of data. Mean surface temperature was given as the independent variable and the data must go into L1.

With all the data appropriately entered into L1 and L2 , a linear regression calculation is performed using the instruction LinReg(a+bx) L1, L2. This will produce the equation as shown in Figure 4.

Figure 4 - Least squares regression line for Question 3b With coefficients written correct to one decimal place, this equation is y = -776.9 + 60.3 x However, we need an equation that relates to the actual data where the variables are not called x and y.

The correct answer for Question 3b was mean duration of warm spell = -776.9 + 60.3 x mean surface temperature The screen shown in Figure 5 shows also shows the values of the correlation coefficient, r, and the coefficient of determination, r?.

Figure 5 - Values of r and r? shown While the values of these coefficients were irrelevant to the examination question discussed here, they may be essential in other circumstances.

Your calculator will only show these coefficients if you have turned on the Diagnostics feature on your calculator found with the CATALOG button. Scroll down and select DiagnosticOn as shown in Figure 6.

Figure 6 - Turning on the Diagnostic function Some other cases where use of a graphics calculator or CAS is helpful are illustrated in the following for in three modules from the 2007 examinations For the Business mathematics module in Examination 1, the TVM Solver was most useful for Question 9.

A loan of $250 000 at 7% pa calculated and repaid monthly is fully paid off in 20 years. Determine the total interest paid over the course of the loan.

All given data is entered into the TVM Solver as shown in Figure 7.

The required monthly repayment of $1938.25 is calculated as shown.

Figure 7 - Data for Business Examination 1 - Question 9 Over the life of the loan, there will be 240 x $1938.25 repayments = $465 180 in total. This total includes the initial loan of $250 000 that had to be repaid.

Therefore, the total interest paid is given by $465 180 - $250 000 = $215 180. This answer is closest to alternative A in the question.

Further use of the TVM Solver was needed in Examination 2 for Questions 2b, 2c, 4b.

In 2007, the matrix functions of a graphics calculator or CAS applied to the Matrices module in: Examination 1 - Questions 3, 6. Examination 2 - Questions 2ci, 2ciii, 2civ, 2cv, 2di, 2dii The sequence mode of entry could also have been usefully applied to Question 4 in the Number patterns module in Examination 2 Well before the November examinations, you must be familiar with your calculator or CAS. If you borrow one at the last moment, it may not be set up in the fashion with which you are familiar.

Use the same calculator in all your class and revision work and replace the batteries before starting your final practice examination. That way, you may be able to avoid problems that may arise from any temporary power loss before the 2008 examinations.

For the November examinations, you may bring in a scientific calculator as well as an approved graphics calculator or CAS. If nothing else, this may be a good insurance policy in case the graphics calculator or CAS should misbehave.

Rob Vermay is a VCE assessor.

© 2008 The Age

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