Fitting, predicting and interpreting models | Year 11

01 / LEARN

The idea, step by step

Choose and fit a rule

Inspect equal-interval data for constant differences or ratios. Two observations can define a line or determine a positive exponential factor across a time gap. Additional observations can then test the proposed relationship.

A fitted rule is not established merely because it passes through two points. Report variables, units, the observed input range and whether predictions are approximate. Keep enough coefficient precision for later calculations.

Residuals show what is missed

A residual equals observed output minus predicted output at the same input. Positive means the observation lies above the prediction; negative means below. Calculate more than one residual when evaluating a data set.

A pattern such as increasingly positive residuals can suggest the wrong growth mechanism. Small irregular errors can support a model over a limited range, but do not justify unlimited extrapolation.

residual = observed - predicted

A common trap

Taking predicted minus observed and reversing the residual sign.

Instead: Write observed minus predicted explicitly before subtracting.

02 / WORKED EXAMPLES

See the thinking

Example 1

A line and a new measurement

Data are $(0, 40), (2, 62), (4, 84)$ . Fit a line, predict at 6, then find the residual if the new observation is 104.

A 2-unit input step adds 22, so gradient is 11.
The rule is $y = 40 + 11 t$ and predicts 106 at 6.
Residual is $104 - 106 = - 2$ .

y = 40 + 11 t

; prediction 106; residual −2.

Example 2

Recover an exponential factor

A quantity is 150 at time 0 and 259.2 at time 3. Fit $P = A b^{t}$ and predict at 5.

$A = 150$ ; then $b^{3} = 259.2 / 150 = 1.728$ .
The positive factor is $b = 1.2$ .
At 5, $P = 150 {(1.2)}^{5} = 373.248$ .

P = 150 {(1.2)}^{t}

; prediction 373.248.

Check your understanding

A model predicts 72 and the observation is 75. Find the residual.

Show answer

$75 - 72 = 3$ , so the observation lies 3 units above the prediction.

03 / WORKSHEET

Your turn

6 questions · 13 marks · Work at your own pace.

Worksheet PDF Worked solutions

Read the questions online

Fit a line to $(0, 12), (3, 27), (6, 42)$ .
2 marks
A prediction is 80 and the observation is 77. Find the residual.
2 marks
Fit $P = A b^{n}$ to $P (0) = 200$ and $P (2) = 288$ .
2 marks
For $y = 7 + 4 x$ , observations at $x = 1, 2, 3$ are 12,14,20. Find residuals.
2 marks
Does accurate fitting between times 2 and 8 alone justify a prediction at 80?
2 marks
An exponential quantity is 60 at time 1 and 135 at time 3. Find its positive factor and value at zero.
3 marks

Read the worked solutions

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

Fit a line to $(0, 12), (3, 27), (6, 42)$ .
2 marks
Each 3-unit input step adds 15, so gradient is 5.
$y = 12 + 5 x$ .
A prediction is 80 and the observation is 77. Find the residual.
2 marks
$77 - 80 = - 3$ .
−3.
Fit $P = A b^{n}$ to $P (0) = 200$ and $P (2) = 288$ .
2 marks
$b^{2} = 288 / 200 = 1.44$ ; use positive factor 1.2.
$P = 200 {(1.2)}^{n}$ .
For $y = 7 + 4 x$ , observations at $x = 1, 2, 3$ are 12,14,20. Find residuals.
2 marks
Predictions are 11,15,19.
Subtract each prediction from its matching observation.
1, −1, 1.
Does accurate fitting between times 2 and 8 alone justify a prediction at 80?
2 marks
80 is far beyond the observed range.
The process may change; local fit does not establish long-range validity.
No.
An exponential quantity is 60 at time 1 and 135 at time 3. Find its positive factor and value at zero.
3 marks
The two-step ratio is 2.25, so $b = 1.5$ .
Initial value $= 60 / 1.5 = 40$ .
Factor 1.5; initial value 40.

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Limitations of extrapolation

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Curriculum reference & lesson scope

SACE Board of South Australia

Stage 1 General Mathematics: algebraic and graphical modelling of linear/exponential change, with interpretation and evaluation.

Official curriculum source ↗

Back to Linear and exponential models