Explanatory/response variables | Year 12

01 / LEARN

Understand the idea.

Start with the purpose of the model

Ask “Which quantity will I know, and which do I want to predict?” The known input is the explanatory variable, usually $x$ ; the predicted quantity is the response, usually $y$ . Table column order does not determine these roles.

An experiment may deliberately change the explanatory variable; an observational study simply records it. In either case, the name describes its role in the model. It does not, by itself, establish that changing it causes the response to change.

Roles affect the regression equation

A least-squares regression of $y$ on $x$ minimises squared errors in the response direction. Predicting $x$ from $y$ changes the response and usually requires a separate fit; rearranging the first equation generally answers a different optimisation problem.

State variable names, measurement units and the population to which the prediction applies. Predicting plant height from age and estimating age from height use the same two measurements, but require different prediction questions and potentially different regression lines.

02 / WORKED EXAMPLES

See the method in action.

EXAMPLE 01

Predict a response

A nursery wants to predict plant height from the number of weeks since planting. Identify the variables.

Worked solution

1
Identify the information available for prediction.
$x = weeks since planting$
The nursery uses the plant's age as the input.
2
Identify the quantity to be predicted.
$y = plant height in cm$
Height is the outcome the nursery wants the model to estimate.
3
State the variable roles.
Age in weeks is explanatory; height in centimetres is the response. These roles come from the nursery's stated purpose, not a causal conclusion from a scatter plot.

Answer

Explanatory: age in weeks. Response: height in cm.

EXAMPLE 02

Reverse the question carefully

A business models sales from advertising cost. A manager instead wants to estimate advertising cost from a target sales level. What changes?

Worked solution

1
State the original prediction direction.
$advertising cost \to sales$
The original model uses cost as its input and minimises prediction errors in sales.
2
State the manager's new prediction direction.
$target sales \to advertising cost$
The known quantity is now sales, and the required response is cost.
3
Refit using cost as the response.
With imperfect association, rearranging a sales-on-cost line does not generally minimise squared cost errors. Fit the regression for the reversed roles using the paired data.

Answer

The response becomes advertising cost, and a new regression is needed.

03 / YOUR TURN

Try it yourself.

Check your understanding

Predict delivery cost using parcel mass. Which variable is the response?

Show answer & reasoning

Delivery cost.

Delivery cost is the response because it is the quantity to be predicted. Parcel mass is the explanatory input used to make that prediction.

Keep practising

6 worksheet questions · 18 marks

Open worksheet Worked solutions

Read the questions online

A sports analyst predicts sprint time using training hours. Name $x$ and $y$ with suitable units.
3 marks
A model predicts electricity use from daily maximum temperature. Which measurement goes on the horizontal axis?
3 marks
A table lists test score first and revision time second. Does this force test score to be $x$ ?
3 marks
A company predicts the time to fill a tank using the flow rate. Identify the response and explain why a negative association may occur.
3 marks
A scientist predicts adult mass from body length. Explain why reversing a fitted regression may require a new fit.
3 marks
Give an observational study in which $x$ is a useful predictor but cannot be claimed to cause $y$ from the study alone.
3 marks

Read the worked solutions

Equivalent correct methods earn the corresponding marks. Include units where asked.

A sports analyst predicts sprint time using training hours. Name $x$ and $y$ with suitable units.
3 marks
The input is training amount.
The response is the predicted time.
$x = training hours per week$ ; $y = sprint time in seconds$ .
A model predicts electricity use from daily maximum temperature. Which measurement goes on the horizontal axis?
3 marks
It is the explanatory variable.
State the temperature unit on the axis.
Daily maximum temperature, in degrees Celsius.
A table lists test score first and revision time second. Does this force test score to be $x$ ?
3 marks
Variable roles follow the modelling question.
No. For predicting score from revision time, time is $x$ regardless of column order.
A company predicts the time to fill a tank using the flow rate. Identify the response and explain why a negative association may occur.
3 marks
The predicted quantity is time.
With fixed volume, $time = \frac{volume}{flow rate}$ , which decreases as flow increases.
Response: filling time. Higher flow rate tends to reduce time for a fixed tank volume.
A scientist predicts adult mass from body length. Explain why reversing a fitted regression may require a new fit.
3 marks
Mass-on-length minimises vertical mass residuals.
Length-on-mass minimises length residuals, not the same objective.
The two fits minimise errors in different response directions.
Give an observational study in which $x$ is a useful predictor but cannot be claimed to cause $y$ from the study alone.
3 marks
Rainy weather can increase both.
Predictive association does not establish that umbrella sales cause raincoat sales.
For example, umbrella sales predicting raincoat sales.

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 2 General Mathematics: statistical models. Original lessons and assessment practice; not an official SACE assessment.

Official curriculum source ↗

In this lesson

Understand the idea.

Start with the purpose of the model

Roles affect the regression equation

See the method in action.

Predict a response

Identify the information available for prediction.

Identify the quantity to be predicted.

State the variable roles.

Reverse the question carefully

State the original prediction direction.

State the manager's new prediction direction.

Refit using cost as the response.

Try it yourself.

Check your understanding

Keep practising