Least-squares regression and prediction (QCE General Mathematics Unit 3)

What this dot point is asking

Once a scatterplot shows a linear association between two numerical variables, you model it with a straight line and use the line to predict. The QCAA General Mathematics 2025 syllabus (Unit 3, Topic 2, Bivariate data analysis 2, "Fitting a linear model to numerical data") asks you to

• fit a least-squares line $y = mx + c$ using technology (your calculator's statistics mode),
• understand and use $m = r\dfrac{s_y}{s_x}$ and $c = \bar{y} - m\bar{x}$ to find the line from summary statistics,
• interpret the slope and $y$-intercept in context,
• distinguish between interpolation and extrapolation,
• use the line to make predictions, and
• recognise and explain the dangers of extrapolation.

Residual plots, the other part of this sub-topic, have their own page, and so does association and causation. In the external assessment you have a scientific calculator, which can run a linear regression from raw data in its statistics mode.

The answer

Explanatory and response variables

The explanatory variable ($x$, horizontal axis) is the one you think explains or predicts the other; the response variable ($y$, vertical axis) is the one being predicted. Hours of training explains run time, temperature explains ice-cream sales, age explains height. Getting this the right way round matters: regressing $y$ on $x$ gives a different line from regressing $x$ on $y$.

The least-squares line

For any line drawn through a scatterplot, each point has a residual: the vertical distance from the point to the line, $y - \hat{y}$ (observed minus predicted). The least-squares line is the one line that makes the sum of the squared residuals as small as possible. In QCAA notation it is written

where $m$ is the slope (gradient) and $c$ is the $y$-intercept. Always write it with the variable names in context, for example $\text{time} = -0.655 \times \text{hours} + 32.69$.

Using the calculator

Put the calculator into its statistics (two-variable or linear regression) mode, enter the $x$ values in one column and the matching $y$ values in the other, and read off the regression coefficients. Two cautions:

• Many scientific calculators write the line as $y = a + bx$ or $y = A + Bx$, so the letter called $a$ or $A$ is the intercept and $b$ or $B$ is the slope. Check which is which before you copy them into $y = mx + c$.
• The same screen gives $r$. Write it down: you need it to judge how good the model is.

Round the coefficients sensibly (usually to 3 or 4 significant figures) and keep more precision in any follow-up calculation.

From summary statistics

When you are given the means, standard deviations and $r$ instead of the raw data, use

where $r$ is the correlation coefficient, $s_x$ and $s_y$ are the (sample) standard deviations of the $x$ and $y$ values, and $\bar{x}$ and $\bar{y}$ are the means. Three facts follow from these formulas:

• The slope has the same sign as $r$ (standard deviations are positive).
• The line always passes through the mean point $(\bar{x}, \bar{y})$, because rearranging $c = \bar{y} - m\bar{x}$ gives $\bar{y} = m\bar{x} + c$.
• A weaker correlation gives a flatter line for the same spreads: the slope is "shrunk" by the factor $r$.

Interpreting the slope and intercept

Use a template and put the context in:

• Slope. "For each increase of 1 [unit of $x$] in [$x$ variable], the [$y$ variable] is predicted to increase (or decrease) by [$|m|$ units of $y$]."
• $y$-intercept. "When [$x$ variable] is 0, the predicted [$y$ variable] is [$c$ units]." Then judge it: if $x = 0$ is far outside the data or is impossible in context, say the intercept has no practical meaning.

Words like "predicted", "on average" or "is associated with" matter: the line describes a trend, not a law, and it does not prove that $x$ causes $y$.

Making predictions

Substitute the $x$ value into the equation. For the run-time model $y = -0.655x + 32.69$, 7 hours of training gives $y = -0.655 \times 7 + 32.69 \approx 28.1$ minutes.

Interpolation and extrapolation

• Interpolation is predicting for an $x$ value within the range of the data. It is generally reliable, especially when the association is strong.
• Extrapolation is predicting for an $x$ value outside the range of the data. It assumes the linear pattern continues where there is no data to support it.

The dangers of extrapolation

Extrapolation can go badly wrong because:

• The pattern may change. Relationships that look linear over a small range often curve, level off or reverse outside it. Training improves run times, but not without limit.
• The prediction can be impossible. The run-time line predicts a 5 km time of about 3.2 minutes at 45 hours of training a week, and a negative time beyond about 50 hours.
• Conditions change over time. Forecasting a future year from past trends (time-series extrapolation) assumes nothing about the situation changes.

Extrapolating a little beyond the data (for example one year ahead in a steady trend) is common and can be reasonable, but you must always label it as extrapolation and say it is less reliable.

How reliable is a prediction?

A prediction is more trustworthy when

1. it is an interpolation,
2. the association is strong ($r$ close to $\pm 1$, so $R^2$ is close to 1: a high proportion of the variation in $y$ is explained by $x$),
3. the residual plot shows no pattern, so a straight line is an appropriate model, and
4. the data set is reasonably large and contains no influential outliers.

Coding time

When $x$ is a year, code it as "years since" a starting year (for example $x = 1$ for 2019). Using 2019 directly gives a huge negative intercept that is hard to interpret and easy to mis-key. State your coding in words; the marker needs it to follow your intercept and your forecast.