The idea, step by step
Start with the purpose of the model
The explanatory variable is used to explain or predict variation in the response. It is normally written x. The response is normally y. These roles come from the modelling question, not from whichever column appears first in a table.
In an experiment, the explanatory variable may be deliberately changed. In an observational study it is simply observed. Calling a variable explanatory does not prove that it causes the response.
Roles affect the regression equation
A regression of y on x minimises vertical response errors. Reversing the question usually requires a different fitted equation; solving the first regression equation for x does not generally give the least-squares regression of x on y.
Record units and the intended population. A model predicting mass from length answers a different practical question from a model predicting length from mass.
A common trap
The explanatory variable must be a proven cause.
Instead: Explanatory identifies the predictor’s role in the model. Observational association alone does not establish cause.
See the thinking
Predict a response
A nursery wants to predict plant height from the number of weeks since planting. Identify the variables.
- The known input for a prediction is weeks since planting.
- The predicted measurement is height.
- Use x=weeks and y=height in cm.
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?
- The original question uses advertising cost as x and sales as y.
- The new prediction reverses the roles.
- Refit with cost as response; simply rearranging an imperfect original fit is not generally least squares for the new purpose.
Check your understanding
Predict delivery cost using parcel mass. Which variable is the response?
Show answer
Delivery cost.
Cost is the quantity being predicted; mass supplies the input.
Your turn
6 questions · 18 marks · Work at your own pace.
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=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.
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 ↗