2023 Paper 2, Q6
The table shows the average superannuation account balance for workers of various ages in two different industries. The coefficient of determination, \(R^2\), for age versus account balance is 0.95 for industry A and 0.96 for industry B. 40-year-old Leigh works in the industry for which age explains a higher percentage of the account balance variation. Tony is 10 years older than Leigh and works in the other industry.
| Age (years) | Industry A ($) | Industry B ($) |
|---|---|---|
| \(22\) | \(7\,500\) | \(8\,100\) |
| \(32\) | \(42\,000\) | \(60\,000\) |
| \(42\) | \(98\,000\) | \(120\,000\) |
| \(52\) | \(160\,000\) | \(210\,000\) |
| \(62\) | \(290\,000\) | \(360\,000\) |
| \(72\) | \(400\,000\) | \(480\,000\) |
Use linear models to predict the difference in current superannuation account balances for Leigh and Tony.
The coefficient of determination picks the industry
Nobody tells you which industry Leigh is in. You are told how to work it out: \(R^2\) is the fraction of the variation in balance that age explains, so the higher value decides.
Decode the people first
\(R^2=0.96\) means age explains 96% of the variation in balances for industry B, against 95% for A. So Leigh is in industry B.
Tony is \(40+10=50\) and works in the other industry, which is A. Write both facts down before any model.
Fit one least-squares line to industry A and another to B. Each person goes into their own industry’s model.
What if you swap the industries?
Switch which industry Leigh is in and change Leigh’s age. Tony stays 10 years older and in the other industry. Putting the people in the wrong industries changes the answer a lot.
Which industry is Leigh in?
0.96 is closer to 1 than 0.95 is. A higher \(R^2\) means age explains more of the variation.
Yes. 96% is more than 95%, so industry B is the one where age explains a higher percentage, and Leigh works there.
They are close, but the question defines Leigh’s industry by which value is higher. The difference of 0.01 is enough to decide it.
QCAA marking guide · 7 marks
\(0.95<0.96\), so age explains a higher percentage of the account balance variation for industry B. Leigh works in industry B, and Tony in industry A.
Marker: correctly identifies the dataset for which age explains a higher percentage of the variation. This may be implied by later working.
\[y=7910x-205\,520\]
Marker: correctly determines the linear model for age against balance for industry A, with \(x\) the age and \(y\) the balance.
\[y=9570x-243\,440\]
Marker: correctly determines the linear model for industry B. Follow-through marks are allowed from here on.
\[y=9570(40)-243\,440=139\,360\]
Marker: substitutes \(x=40\) into the appropriate equation and calculates Leigh’s current balance.
\[x=40+10=50,\qquad y=7910(50)-205\,520=189\,980\]
Marker: substitutes \(x=50\) into the appropriate equation and calculates Tony’s current balance.
\[189\,980-139\,360=50\,620\]
The difference in their account balances is predicted to be $50 620.
Marker: calculates the difference in current account balances for Leigh and Tony.
Define \(x\) and \(y\), label each model with its industry, and say who is in which industry before you substitute.
Marker: shows logical organisation, communicating key steps, with connecting statements and correct use of equals signs.
Who has more, and why
Tony has about $50 620 more, even though industry B pays into super faster. The extra 10 years of saving outweighs the steeper line.
With Leigh in A and Tony in B, the predictions are $110 880 and $235 060, a difference of $124 180. That is why the first mark is for choosing the industry.
Fitting two least-squares lines and substituting into them is routine. The unfamiliar part is that the question hides who is in which industry behind \(R^2\), a statistic you usually just report. You have to read it as "the percentage of variation explained", decide that 96% beats 95%, and then follow the clue about Tony to the other industry. I want you to write the two people, their ages and their industries as a list before you touch your calculator.
Question wording and marking-guide steps are from the 2023 QCAA General Mathematics external assessment, © State of Queensland (QCAA) 2023, licensed under CC BY 4.0, and have been adapted. Tangent Tuition is not affiliated with the QCAA.
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