2022 Paper 2, Q3
In a company’s first 10 years of operation, the average annual profit \((\bar y)\) was $9660 with a standard deviation \((s_y)\) of $3010. Fitting a least-squares line to the data comparing annual profit \((y)\) to the year of operation \((x)\) produced a correlation coefficient of 0.9987.
Show that the predicted profit, to the nearest dollar, for this company in the 11th year of operation will be $15 121.
The years give you the missing statistics
You are not given any data, but you know exactly what \(x\) is: the years 1 to 10. That gives you \(\bar x\) and \(s_x\). With those, the slope formula and the mean point build the whole line.
Two formula-sheet rules in a row
Put the ten years into your calculator as a list. It gives \(\bar x=5.5\) and \(s_x=3.02765\). That first mark is the one most people miss.
\(b=r\dfrac{s_y}{s_x}\), then \(a=\bar y-b\bar x\). Keep at least three decimal places in \(b\), or the final answer drifts off $15 121.
You are given the answer, so the marks are all for the working. Your last line has to land on $15 121 from your own values.
Number the years another way
The guide accepts \(x=0,1,\ldots,9\) as well. Switch between the two. The mean and the intercept change, the slope does not, and the prediction for the 11th year comes out the same.
One standard deviation is missing. Where does it come from?
The question never lists the \(x\)-values, but it tells you what they are: the first 10 years of operation. That is a complete data set.
Yes. Enter 1 to 10 into your calculator and read off \(s_x=3.02765\). This is the hidden step in the question.
\(s_y\) is measured in dollars and \(s_x\) in years, so they cannot be the same thing. You need \(s_x\) from the years themselves.
QCAA marking guide · 5 marks
\[x=1,2,\ldots,10\quad\Rightarrow\quad\bar x=5.5,\qquad s_x=3.02765\]
Marker: correctly determines \(\bar x\) and \(s_x\). Alternative numbering such as \(x=0,1,\ldots,9\) is accepted.
\[b=r\frac{s_y}{s_x}=0.9987\times\frac{3010}{3.02765}=992.878\]
Marker: determines \(b\). This mark may be implied by later working.
\[a=\bar y-b\bar x=9660-992.878\times5.5=4199.17\]
Marker: determines \(a\). Follow-through marks are allowed for \(b\), \(a\) and the prediction.
\[y=a+bx=4199.17+992.878(11)=15\,120.83\]
The predicted profit in the 11th year is $15 121.
Marker: determines the 11th year profit to the nearest dollar.
Head the working "\(x\) parameters", "Given" and "Least-squares line parameters" as the guide does, and write each formula before you substitute.
Marker: shows logical organisation, communicating key steps, with connecting statements and correct use of equals signs.
Two things that can go wrong
Your calculator shows two standard deviations. \(s_x=3.02765\) is the sample one. The other, 2.87228, gives \(b=1046.6\) and a prediction of $15 416, which does not match the question.
Rounding \(s_x\) to 3 makes \(b=1002.0\) and the prediction $15 171. A "show that" question gives you a target, so if you miss it, check your rounding first.
Building a least-squares line from summary statistics is a standard skill. This question leaves out half of the summary. You have to see that "year of operation" for the first 10 years is a known list of numbers, and get \(\bar x\) and \(s_x\) from it yourself. When a formula needs a value the question does not give, I want you to ask whether the context already tells you the data behind it.
Question wording and marking-guide steps are from the 2022 QCAA General Mathematics external assessment, © State of Queensland (QCAA) 2022, licensed under CC BY 4.0, and have been adapted. Tangent Tuition is not affiliated with the QCAA.
Spot a mistake on this page? Report it