Fitting a linear model

Covers AS91581

The line of best fit

When the pattern is roughly linear, summarise it with a straight-line model (a linear regression):

$y = a + bx$

$b$ — the slope (gradient): the change in $y$ for each 1-unit increase in $x$ .
$a$ — the intercept: the predicted $y$ when $x = 0$ .
Statistics software fits the line to sit as close as possible to all the points; your job is to interpret it.

Scatter plot of plant height against fertiliser with a straight trend line drawn through the points, sloping upward.

Interpreting the slope and intercept in context

Slope in context: "for each extra 1 unit of $x$ , the [response] increases (or decreases) by about $b$ [units of $y$ ], on average."
Intercept in context: the predicted response when $x = 0$ — but this is only meaningful if $x = 0$ is within or near the data.
Always attach units and say "on average" — the model is a summary of a trend, not an exact rule.

Worked ExampleInterpret a fitted model

A linear model for the plant data is $y = 12 + 0.8x$ , where $x$ is fertiliser (g) and $y$ is height (cm). Interpret the slope and the intercept.

Slope

The slope is $0.8$ : for each extra gram of fertiliser, plant height increases by about $0.8$ cm on average.

Intercept

The intercept is $12$ : the model predicts a height of about $12$ cm when no fertiliser is applied ( $x = 0$ ). This is only meaningful if $x = 0$ is close to the measured data.

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

A linear model is $y = 5 + 2x$ . State the slope and the intercept.

Merit

A model for a car's value is $y = 26000 - 2200x$ , where $x$ is the car's age in years and $y$ is its value in dollars. Interpret the slope in context.

Excellence

For the model $y = 26000 - 2200x$ (car value in dollars, age $x$ in years), interpret the intercept and comment on whether it is meaningful.

Fitting a linear model

Covers AS91581

The line of best fit

When the pattern is roughly linear, summarise it with a straight-line model (a linear regression):

$y = a + bx$

$b$ — the slope (gradient): the change in $y$ for each 1-unit increase in $x$ .
$a$ — the intercept: the predicted $y$ when $x = 0$ .
Statistics software fits the line to sit as close as possible to all the points; your job is to interpret it.

Scatter plot of plant height against fertiliser with a straight trend line drawn through the points, sloping upward.

Interpreting the slope and intercept in context

Slope in context: "for each extra 1 unit of $x$ , the [response] increases (or decreases) by about $b$ [units of $y$ ], on average."
Intercept in context: the predicted response when $x = 0$ — but this is only meaningful if $x = 0$ is within or near the data.
Always attach units and say "on average" — the model is a summary of a trend, not an exact rule.

Worked ExampleInterpret a fitted model

A linear model for the plant data is $y = 12 + 0.8x$ , where $x$ is fertiliser (g) and $y$ is height (cm). Interpret the slope and the intercept.

Slope

The slope is $0.8$ : for each extra gram of fertiliser, plant height increases by about $0.8$ cm on average.

Intercept

The intercept is $12$ : the model predicts a height of about $12$ cm when no fertiliser is applied ( $x = 0$ ). This is only meaningful if $x = 0$ is close to the measured data.

Test yourself

Practice by grade

One question each at Achieved, Merit and Excellence. Have a go, then compare with the model answer.

Achieved

A linear model is $y = 5 + 2x$ . State the slope and the intercept.

Merit

A model for a car's value is $y = 26000 - 2200x$ , where $x$ is the car's age in years and $y$ is its value in dollars. Interpret the slope in context.

Excellence

For the model $y = 26000 - 2200x$ (car value in dollars, age $x$ in years), interpret the intercept and comment on whether it is meaningful.