Topics

Fitting a model, predicting, and the conclusion

Covers AS91581

Choosing the model

Look at the scatter plot and match the model to the shape. This is the decision, and it comes before any fitting.
- Points scattered about a straight line → linear model
- Points following a curve → non-linear model
The standard explicitly permits non-linear relationships. Forcing a straight line onto curved data is a substantive error, not a simplification.

The linear model

$y = a + bx$

$b$ is the slope — the change in $y$ for a one-unit increase in $x$ .
$a$ is the intercept — the predicted $y$ when $x = 0$ .
Interpreting the slope is where the marks are, and it must be done with both variables' units and the context:
- Not: "the slope is $-0.043$ "
- But: "the slope is $-0.043$ , meaning that for each additional 1 000 km on the odometer, the predicted sale price falls by about $43."
Interpreting the intercept requires care. It is only meaningful if $x = 0$ is within or near the data range and makes sense in context. If your data covers houses of 80–300 m², the intercept describes a house of zero floor area, which is meaningless — say so rather than interpreting it.

Residuals

A residual is the vertical distance from a point to the fitted model:

$\text{residual} = \text{observed } y - \text{predicted } y$

A positive residual means the actual value is above the model; negative means below.
The residual plot is the model's report card. Plot residuals against $x$ and look:

What the residual plot shows	What it means
Random scatter about zero	The model fits the shape of the data
A curved pattern	The relationship is non-linear — wrong model
A fan shape	Scatter changes across the range; predictions are less reliable at one end
An isolated extreme point	An outlier worth investigating

Producing and interpreting a residual plot is one of the clearest ways to evidence "evaluating the adequacy of the model", which is named in the Excellence criterion.

Making a prediction

Substitute your $x$ value into the model and state the result in context, with units.
Then qualify it. Three things must be said:
- The uncertainty. "Given the scatter, actual prices for a car with this odometer reading have ranged from about $8 000 to $16 000, so the prediction of $12 000 should be read as a central estimate rather than an exact figure."
- Whether it is interpolation or extrapolation.
- Whether the scatter is larger at that part of the range.

Interpolation and extrapolation

Interpolation — predicting within the range of the data. Reasonable.
Extrapolation — predicting outside it. Unreliable, and often nonsensical.
Say which you are doing, every time. Extrapolation fails because there is no evidence the pattern continues beyond the data, and a linear model extended far enough will eventually predict impossible values — negative prices, negative times, percentages above 100.

Association is not causation

A fitted model shows that two variables move together. It does not show that one causes the other.
Three alternatives always exist, and naming the relevant one is a contextual judgement:
- A confounding variable influences both.
- The causation runs the other way.
- Chance, particularly in a small data set.
Say this explicitly in the conclusion, and name the specific alternative that applies to your context — not the generic phrase.

The conclusion

A conclusion that scores well:
- Answers the relationship question posed at the start
- Describes the nature and strength with evidence from the data
- Explains the relationship in context
- States the prediction with its uncertainty and limitations
- Evaluates the model — residuals, whether the form was right, where it is least reliable
- Considers other relevant variables and says what each would add
- Notes limitations — the data set's origin, its size, whether the population is narrow

Tips and Tricks

Interpret the slope with units and context. "For each extra square metre of floor area, the predicted price rises by about $3 100" — that sentence is worth more than the fitted equation.

Do not interpret the intercept unless $x = 0$ is meaningful. Saying "the intercept is not meaningful here because a house of zero floor area does not exist" earns more than a forced interpretation.

Produce a residual plot and say what it shows. It is the most direct evidence of model evaluation available to you.

Label every prediction as interpolation or extrapolation, and give a realistic range rather than a single number.

Never quote $R^2$ . Describe the strength from the scatter, in words, and say what it means for prediction accuracy.

In the conclusion, name the specific confounding variable you are worried about. "Correlation is not causation" as a standalone sentence earns nothing.

Worked ExampleFitting, predicting and concluding

A student investigating house floor area (m²) against sale price ($000s) for 120 sales in one New Zealand suburb has fitted:

$\text{price} = 285 + 3.1 \times \text{floor area}$

The data cover floor areas from 75 m² to 240 m². At a floor area of 150 m², observed prices in the data range from about $620 000 to $920 000. The residual plot shows random scatter but with the spread widening as floor area increases.

Write the model interpretation, a prediction for a 160 m² house, and the conclusion.

Step 1 — Interpret the slope

$b = 3.1$

Step 2 — Interpret the intercept, carefully

$a = 285$

Step 3 — Make the prediction

For a floor area of 160 m²:

$\text{price} = 285 + 3.1 \times 160 = 285 + 496 = 781$

Step 4 — Qualify the prediction properly

Interpolation. 160 m² lies within the data range of 75–240 m², so this is interpolation and is reasonable on that count.

Uncertainty. The prediction should not be read as a precise figure. At a floor area of 150 m² — close to the value being predicted — actual sale prices in the data ranged from about $620 000 to $920 000, a spread of $300 000. The predicted $781 000 is a central estimate, and an individual house of this size could reasonably sell anywhere within a range of that order.

Reliability varies across the range. The residual plot shows the spread widening as floor area increases, so predictions for large houses are less reliable than for small ones. At 160 m² we are in the middle of the range, where reliability is moderate.

Why the scatter is so large in this context. Floor area is only one determinant of house price. Two houses of identical size in the same suburb can differ greatly in price because of condition and age, land area, the number of bathrooms, renovations, sun and outlook, off-street parking, and which street they are on. The model captures none of these.

Step 5 — Evaluate the model

Step 6 — The conclusion

Answering the question. This investigation examined the relationship between floor area and sale price for 120 house sales in this suburb.

Nature and strength. There is a positive, linear relationship between floor area and sale price: larger houses sell for more. The relationship is moderate in strength — the upward pattern is clear, but at a given floor area prices vary by around $300 000, so floor area alone does not determine price.

Explanation in context. The positive relationship is expected: floor area is a direct measure of the accommodation a buyer receives, and more space commands a higher price. The relationship is not stronger because buyers value many attributes other than size, and because this suburb contains a mix of house ages and conditions.

Prediction. A 160 m² house in this suburb is predicted to sell for about $781 000, with substantial uncertainty — a range of a few hundred thousand dollars around that figure would be consistent with the data.

Model evaluation. A linear model fits the form of the relationship well, with no curvature in the residuals. Its main weakness is that the scatter increases with floor area, so predictions for large houses are less reliable.

Other relevant variables. Three additions would substantially improve the model:

Land area, since a significant part of a property's value is in the section rather than the dwelling — this is also consistent with the large positive intercept.
Age or condition of the house, which would explain much of the price variation between houses of the same size.
Date of sale, since the data cover a period during which the market itself may have moved; without this, market movement is being absorbed into the scatter.

Association, not causation. The model shows that floor area and price move together; it does not establish that adding floor area to an existing house would raise its value by $3 100 per square metre. The most likely reason for caution is confounding: larger houses in this suburb also tend to be on larger sections, in better condition, and with more bathrooms, so the slope of 3.1 is capturing the combined effect of size and everything correlated with it — not the effect of size alone.

Limitations. The findings apply to this suburb over this period only and should not be generalised to other suburbs or to other market conditions. The data set of 120 sales is adequate for describing the relationship but does not support conclusions about individual properties.

Why this reaches Excellence

Fitting a model, predicting, and the conclusion

Covers AS91581

Choosing the model

Look at the scatter plot and match the model to the shape. This is the decision, and it comes before any fitting.
- Points scattered about a straight line → linear model
- Points following a curve → non-linear model
The standard explicitly permits non-linear relationships. Forcing a straight line onto curved data is a substantive error, not a simplification.

The linear model

$y = a + bx$

$b$ is the slope — the change in $y$ for a one-unit increase in $x$ .
$a$ is the intercept — the predicted $y$ when $x = 0$ .
Interpreting the slope is where the marks are, and it must be done with both variables' units and the context:
- Not: "the slope is $-0.043$ "
- But: "the slope is $-0.043$ , meaning that for each additional 1 000 km on the odometer, the predicted sale price falls by about $43."
Interpreting the intercept requires care. It is only meaningful if $x = 0$ is within or near the data range and makes sense in context. If your data covers houses of 80–300 m², the intercept describes a house of zero floor area, which is meaningless — say so rather than interpreting it.

Residuals

A residual is the vertical distance from a point to the fitted model:

$\text{residual} = \text{observed } y - \text{predicted } y$

A positive residual means the actual value is above the model; negative means below.
The residual plot is the model's report card. Plot residuals against $x$ and look:

What the residual plot shows	What it means
Random scatter about zero	The model fits the shape of the data
A curved pattern	The relationship is non-linear — wrong model
A fan shape	Scatter changes across the range; predictions are less reliable at one end
An isolated extreme point	An outlier worth investigating

Producing and interpreting a residual plot is one of the clearest ways to evidence "evaluating the adequacy of the model", which is named in the Excellence criterion.

Making a prediction

Substitute your $x$ value into the model and state the result in context, with units.
Then qualify it. Three things must be said:
- The uncertainty. "Given the scatter, actual prices for a car with this odometer reading have ranged from about $8 000 to $16 000, so the prediction of $12 000 should be read as a central estimate rather than an exact figure."
- Whether it is interpolation or extrapolation.
- Whether the scatter is larger at that part of the range.

Interpolation and extrapolation

Interpolation — predicting within the range of the data. Reasonable.
Extrapolation — predicting outside it. Unreliable, and often nonsensical.
Say which you are doing, every time. Extrapolation fails because there is no evidence the pattern continues beyond the data, and a linear model extended far enough will eventually predict impossible values — negative prices, negative times, percentages above 100.

Association is not causation

A fitted model shows that two variables move together. It does not show that one causes the other.
Three alternatives always exist, and naming the relevant one is a contextual judgement:
- A confounding variable influences both.
- The causation runs the other way.
- Chance, particularly in a small data set.
Say this explicitly in the conclusion, and name the specific alternative that applies to your context — not the generic phrase.

The conclusion

A conclusion that scores well:
- Answers the relationship question posed at the start
- Describes the nature and strength with evidence from the data
- Explains the relationship in context
- States the prediction with its uncertainty and limitations
- Evaluates the model — residuals, whether the form was right, where it is least reliable
- Considers other relevant variables and says what each would add
- Notes limitations — the data set's origin, its size, whether the population is narrow

Tips and Tricks

Interpret the slope with units and context. "For each extra square metre of floor area, the predicted price rises by about $3 100" — that sentence is worth more than the fitted equation.

Produce a residual plot and say what it shows. It is the most direct evidence of model evaluation available to you.

Label every prediction as interpolation or extrapolation, and give a realistic range rather than a single number.

Never quote $R^2$ . Describe the strength from the scatter, in words, and say what it means for prediction accuracy.

In the conclusion, name the specific confounding variable you are worried about. "Correlation is not causation" as a standalone sentence earns nothing.

Worked ExampleFitting, predicting and concluding

A student investigating house floor area (m²) against sale price ($000s) for 120 sales in one New Zealand suburb has fitted:

$\text{price} = 285 + 3.1 \times \text{floor area}$

Write the model interpretation, a prediction for a 160 m² house, and the conclusion.

Step 1 — Interpret the slope

$b = 3.1$

Step 2 — Interpret the intercept, carefully

$a = 285$

Step 3 — Make the prediction

For a floor area of 160 m²:

$\text{price} = 285 + 3.1 \times 160 = 285 + 496 = 781$

Step 4 — Qualify the prediction properly

Interpolation. 160 m² lies within the data range of 75–240 m², so this is interpolation and is reasonable on that count.

Step 5 — Evaluate the model

Step 6 — The conclusion

Answering the question. This investigation examined the relationship between floor area and sale price for 120 house sales in this suburb.

Other relevant variables. Three additions would substantially improve the model:

Land area, since a significant part of a property's value is in the section rather than the dwelling — this is also consistent with the large positive intercept.
Age or condition of the house, which would explain much of the price variation between houses of the same size.
Date of sale, since the data cover a period during which the market itself may have moved; without this, market movement is being absorbed into the scatter.