Find and use the equation of the least-squares regression line to model a linear relationship between two variables

What this dot point is asking

NESA wants you to find the equation of the least-squares regression line using calculator statistics functions, write it in $y = mx + b$ form, use it to predict $y$ from $x$, and interpret the gradient and intercept in the context of the worded problem.

The answer

The least-squares regression line

For bivariate data, the least-squares regression line is the straight line that minimises the sum of the squared vertical distances from the data points to the line. It is the standard "best-fit" line for linear association.

The equation has the form:

where $m$ is the gradient and $b$ is the $y$-intercept.

Finding the line on a calculator

You will not be asked to compute the gradient and intercept by hand. The procedure on a NESA-approved scientific calculator:

1. Enter statistics mode (typically MODE STAT 2-VAR or similar).
2. Enter the $(x, y)$ pairs into the statistical lists.
3. Read off $m$ (sometimes labelled $a$ or B) and $b$ (sometimes labelled $A$ or $a$) from the regression-result menu.
4. Read off $r$ (correlation coefficient) at the same time.

Different calculator models label these differently. Practise on the exact model you will use in the exam.

Predicting $y$ from IMATH_18

Substitute the $x$ value into the line equation. This is the model's predicted $y$ at that $x$. The actual value may be slightly different; the line is the best linear fit, not a guarantee.

Interpreting the gradient

The gradient $m$ has units of (y units) per (x unit). For every unit increase in $x$, $y$ changes by $m$ units on average.

Always include the word "average" or "on average" and the units in your answer. Markers reward this explicitly.

Interpreting the $y$-intercept

The intercept $b$ is the predicted $y$ value when $x = 0$. In context this is sometimes meaningful (e.g. base salary at zero years of experience) and sometimes extrapolation (e.g. predicted food spending at zero income).

If $x = 0$ lies well outside the dataset, comment that the intercept is an extrapolation and may be unreliable.

When to use the line

The least-squares line is appropriate when:

• The scatterplot suggests an approximately linear relationship.
• The correlation coefficient $|r|$ is moderately strong or stronger.
• There are no extreme outliers distorting the fit.

If the scatterplot is clearly non-linear, the line will be a poor model even if $r$ is not zero.