The idea, step by step
Translate the slope into a rate
The slope gives the model’s predicted change in the response for a one-unit increase in the explanatory variable. Its unit is response units per explanatory unit. A negative slope predicts a decrease. It does not promise that each observed individual changes by exactly that amount.
If a model uses hours, changing the input to minutes changes the numerical slope. The physical rate must still describe the same relationship.
Check whether the intercept has practical meaning
The intercept is the predicted response when x=0. If zero lies well outside the observed data, or the situation cannot have x=0, the intercept may have no useful direct interpretation. It is still part of the fitted equation.
A sensible interpretation mentions the model and units. Avoid claiming that the intercept is an observed value or that an extrapolated prediction must be possible.
A common trap
The intercept is always a real measured starting value.
Instead: It is a fitted value at x=0 and may lie outside the meaningful data range.
See the thinking
A delivery model
Delivery cost C dollars is modelled by C=6.5+1.8m, where m is parcel mass in kg. Interpret both coefficients.
- At m=0, the model predicts a cost of 6.50 dollars.
- Each extra kg increases predicted cost by 1.80 dollars.
- A zero-mass parcel may not be an actual service, so confirm the applicable mass range.
A different time unit
A model is d=12+30t, with d in km and t in hours. Rewrite it using u minutes.
- Since t=u/60, replace t by u/60.
- d=12+30(u/60)=12+0.5u.
- The rate is 0.5 km/min, equivalent to 30 km/h.
Check your understanding
A fitted line predicts fuel remaining F=50-0.08d, with d in km. Interpret the slope.
Show answer
Predicted fuel falls by 0.08 L for each additional km.
The slope has units litres per kilometre and is negative.
Your turn
6 questions · 18 marks · Work at your own pace.
Read the questions online
- A fitted crop yield is Y=2+0.4R, with Y in tonnes and R in mm rainfall. Interpret 0.4.3 marks
- A model T=90-2.5x predicts test time in minutes from weeks of training. Interpret -2.5.3 marks
- A tree model h=1.2+0.7t was fitted only for ages 5–12 years. Is 1.2 m a reliable measured height at birth?3 marks
- Change y=4+12x from x hours to t minutes.3 marks
- The model y=5+0.03x uses x in dollars. Express it with z in hundreds of dollars.3 marks
- A regression gives an intercept of -8 for mass predicted from length. Must the entire model be useless?3 marks
Read the worked solutions
Equivalent correct methods earn the corresponding marks. Include units where asked.
- A fitted crop yield is Y=2+0.4R, with Y in tonnes and R in mm rainfall. Interpret 0.4.3 marks
State response units per explanatory unit.
Qualify the statement as a model prediction.
The model predicts 0.4 more tonnes for each additional mm of rainfall. - A model T=90-2.5x predicts test time in minutes from weeks of training. Interpret -2.5.3 marks
The negative sign indicates a decrease.
Each extra training week predicts a 2.5-minute decrease in test time. - A tree model h=1.2+0.7t was fitted only for ages 5–12 years. Is 1.2 m a reliable measured height at birth?3 marks
Zero is outside the observed age range.
No. It is the extrapolated prediction at age zero, not a measured birth height. - Change y=4+12x from x hours to t minutes.3 marks
Substitute x=t/60.
12/60=0.2.
y=4+0.2t. - The model y=5+0.03x uses x in dollars. Express it with z in hundreds of dollars.3 marks
x=100z.
0.03×100=3.
y=5+3z. - A regression gives an intercept of -8 for mass predicted from length. Must the entire model be useless?3 marks
A negative extrapolated zero-length prediction does not alone invalidate a local fitted relationship.
No. The intercept may be outside the relevant length range; inspect fit and predictions within that range.
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 ↗