01 · Substitute
A fitted model is ŷ = 4 + 1.8x. Predict y for x = 5.
Hint
Include the intercept.
Worked solution
ŷ = 13, in the response’s units.
Understand · explore · practise
Regression predictions and reliability
Calculate fitted predictions, classify interpolation using the explanatory-variable range and judge extrapolation, scatter, sample context and model limitations.
Before you startRegression equations, scatter diagrams and interpreting coefficients.
01 / Calculate, then assess
A numerical answer still needs a reliability judgement.
Check the supplied x against the observed explanatory-variable range.
Interpolation uses x values inside that range; extrapolation extends outside it.
The formula may accept a number even when the prediction is poorly supported. Keep the arithmetic and the judgement separate. A value inside the range is usually better supported by nearby observations, but it is not guaranteed accurate.
Check the explanatory rangeExplore
Constructed delivery pairs (km, minutes): (2,14), (4,16), (6,24), (8,29), (10,31), (12,39). Their fitted line is t = 8 + 2.5d. The observed distance range is 2–12 km.
At 0 km, predicted duration is 8 minutes. This is extrapolation beyond the observed distance range, even though the formula can be evaluated.
02 / Use the fitted direction and units
Predict the response from the supplied explanatory value.
For t = 8 + 2.5d, predict duration at d = 8 km.Worked example
t̂ = 8 + 2.5×8 = 28 minutes
Use distance in the unit used when fitting.
The observed response at 8 km in the constructed data is 29
Prediction and observation can differ.
Residual = 29 − 28 = 1 minute
A fitted prediction is not a rewritten measurement.
02 · Unit mismatch
The delivery model uses kilometres. A distance is supplied as 8000 metres. What should be substituted?
Hint
Convert the explanatory value first.
Worked solution
d = 8 km, giving 28 minutes. Substituting 8000 into the kilometre equation is a unit error.
03 / Use the explanatory range
Compare x with x, not y with y.
Watch: cross the observed distance boundary
Pause, replay or seek freely. The notes explain the same idea and stay in view.
03 · Classify
Observed x values range from 2 to 12. Classify predictions at x = 6 and x = 16.
Hint
Compare the supplied explanatory values with 2 and 12.
Worked solution
At 6: interpolation. At 16: extrapolation. This classification does not require inspecting the predicted y range.
04 · Boundary
A prediction uses x = 12, the largest observed explanatory value. Is it outside the observed x range?
Hint
Equality is still on the range boundary.
Worked solution
No. It is a prediction at the boundary of the observed range. It may still differ from the observed response at that x.
04 / Do not use response range as the test
A fitted value can sit beyond observed y values.
The constructed responses run from 14 to 39 minutes. Predict at d = 2.2 km.Worked example
Predicted duration = 8 + 2.5×2.2 = 13.5 minutes
This is below the smallest observed duration, 14.
But 2.2 km is between 2 and 12 km
The explanatory value is inside the observed range.
It is interpolation, despite being outside the observed response range
Classification and prediction accuracy are separate questions.
05 · Reverse mistake
Observed x values are 10–20 and observed y values are 30–80. A model predicts y = 50 at x = 25. Is it interpolation because 50 lies between 30 and 80?
Hint
The supplied x lies beyond its observed range.
Worked solution
No. It is extrapolation because x = 25 is outside 10–20.
06 · Observation or fit
At d = 2 km, the dataset records 14 minutes while the line predicts 13. Which is the observed response?
Hint
Do not overwrite the measurement with the fitted value.
Worked solution
14 minutes is observed; 13 is predicted. The residual is 1 minute.
05 / Explain why extension is risky
The relationship may change outside the studied range.
07 · Beyond the data
Why is predicting delivery duration at 60 km from data covering only 2–12 km questionable?
Hint
The same straight trend may not continue.
Worked solution
Route types, traffic and other conditions may differ, and there are no observations near 60 km to support the extension. The formula gives a number but the model needs new evidence.
08 · Impossible forever
A decreasing fitted line eventually predicts negative values for a quantity that cannot be negative. What should you conclude?
Hint
Physical context limits a model.
Worked solution
The line cannot remain suitable indefinitely. Restrict its use to a justified domain or choose a supported alternative; do not interpret impossible outputs as measurements.
06 / Assess more than the range
Nearby x values do not cure a poor fit.
09 · Widely scattered sample
A prediction is within the observed x range, but responses vary widely near that x. Is an exact individual prediction well supported?
Hint
Interpolation does not remove residual variation.
Worked solution
No. The fitted value may summarise a central trend, while individual outcomes can differ substantially. State the uncertainty rather than promising an exact value.
10 · New setting
A line fitted to one city’s deliveries is used in a different city at a distance inside the original range. Is the range check enough?
Hint
Context and sampling also matter.
Worked solution
No. Road layout, traffic, sampling and other conditions can differ. Being inside the numerical x range does not establish applicability to a new population.
07 / Write a qualified prediction
Give the number, unit and reasoned limitation.
11 · Suitable wording
The fitted delivery model predicts 28 minutes at 8 km. Give a careful report.
Hint
Say what the model predicts and what the range supports.
Worked solution
The model predicts about 28 minutes for an 8 km delivery. This lies within the observed distance range, but individual times vary and the prediction depends on a comparable setting.
12 · Strong correlation
Does strong correlation make every extrapolation reliable?
Hint
Strength was assessed on the observed data.
Worked solution
No. A tight trend within the studied range can change beyond it. Correlation strength alone does not validate an unsupported extension.
08 / Check range, fit and context
A calculated prediction is not a guarantee.
Substitute the correct units and prediction direction. Classify the input using its explanatory-variable range, then assess scatter, model shape, sampling and context. Keep fitted and observed responses separate.
Section 1 of 8 · Calculate, then assess