Predict from a fitted model: interpolation and extrapolation

To classify a prediction as interpolation or extrapolation, which range must be checked?

The invented reported fit is c=102−0.8t, with observed times 10–50 minutes. Predict which settings use extrapolation and which outputs violate the physical charge range of 0% to 100%. Move the time setting; solid and dashed segments distinguish the observed interval from extensions

At 60 minutes the equation gives 54%. What does the plausible charge value establish?

Predicting from a fitted model

A fitted model predicts the response by substituting an explanatory-variable value into y = a + bx. The prediction is interpolation when that value is within the inclusive observed range and extrapolation when it is outside

1. 1Identify the variables Use x for the explanatory variable and y for the response variable in the fitted model \(\displaystyle x=\text{EV},\quad y=\text{RV}\)
2. 2Substitute the explanatory value Replace x with the stated explanatory value before simplifying the model \(\displaystyle \hat{y}=12+3\left(5\right)=27\)
3. 3Classify the range position A value inside or at an endpoint of the observed explanatory range gives interpolation; a value outside gives extrapolation \(\displaystyle 20\le x\le40\Rightarrow\text{interpolation}\)
4. 4Judge the prediction Interpolation is supported by the observed explanatory values, while extrapolation extends the model beyond those values \(\displaystyle x<20\text{ or }x>40\Rightarrow\text{extrapolation}\)

Given values

\(\displaystyle y=18+2.5x,\quad x=12\)

Calculate the slope contribution in minutes for the given model and input

Given values

\(\displaystyle a=18,\quad bx=30\)

Add the intercept to calculate the predicted travel time in minutes

Using the response value to judge reliability. One of these lines is right. Which one?

Produce a justified prediction report

Use two original examples. A: the fitted card-sorting model ŷ=12/5+(31/35)x, observed allocated times 1–6 minutes. Predict at x=1,3.5,6,7.5 minutes, keeping exact values before reporting to two decimal places. Classify each from the inclusive input range. B: an invented reported battery fit c=102−0.8t, observed times 10–50 minutes and physical charge bounds 0–100%. Predict at t=0,10,30,50,60,130 minutes and classify each

1. 1Explain limits, not only arithmetic Identify outputs violating a physical bound, and plausible outputs that are still extrapolations. Explain why interpolation does not guarantee individual accuracy, model suitability or a causal effect. For A compare the model’s prediction at x=3 with the observed six-card session C; do not relabel a prediction as an observation
2. 2Choose and verify technology Build a table with input, exact prediction, reported prediction, input-range classification, physical check and qualified decision. Record the stored coefficients and formulas. Verify endpoints and at least one outside value independently. Plot A over x=0..13 minutes,y=0..15 cards, with the observed interval marked. Plot B over t=0..140 minutes,c=−20..110%, so impossible extrapolations remain visible
3. 3Write and save before opening the model Save a complete contextual explanation, labelled plots, exact calculations, range decisions and tool record. Independent writing, drawing and external-tool execution are not automatically marked; use self or teacher review

Model response: input support and output plausibility are separate

For A, predictions at 1,3.5,6,7.5 minutes are 23/7,11/2,54/7,633/70 cards, reported as 3.29,5.50,7.71,9.04. The first three inputs are interpolation, including the endpoints 1 and 6; 7.5 is extrapolation. The count prediction 9.04 is plausible but not supported by the observed time range

1. 1Battery results For B, predictions at 0,10,30,50,60,130 minutes are 102%,94%,78%,62%,54%,−2%. Inputs 10,30,50 are interpolation; 0,60,130 are extrapolation. Both 102% and −2% violate physical charge bounds. 54% is physically possible but still extrapolated. Do not silently clamp impossible results to 0% or 100% and treat them as validated predictions
2. 2Prediction is not an observation At x=3 the card model predicts 177/35≈5.057 cards, while session C actually recorded 6. The difference 33/35 cards is its residual. Even an in-range prediction need not match an individual case. Assess the form, residual pattern, unusual cases, sampling and whether conditions remain comparable; later 3D develops residual-pattern checking
3. 3One reproducible workflow Store the exact card coefficients 12/5 and 31/35. In each row calculate a+b*x from the unrounded stored values. Classify with the inclusive test 1≤x≤6, and keep rounding in a separate display column. For B use 102−0.8*t and the independent inclusive test 10≤t≤50; test c<0 or c>100 separately. Save the chosen windows, source ranges and formulas. No external tool execution is established by typing a correct answer here

Given values

\(\displaystyle \hat{y}=12/5+(31/35)x,\quad x=3.5\)

Report the predicted cards at 3.5 minutes to two decimal places

Given values

\(\displaystyle \hat{y}=12/5+(31/35)x,\quad x=7.5\)

Report the prediction at 7.5 minutes to two decimal places

Given values

\(\displaystyle \hat c=102-0.8t,\quad t=130\)

Calculate the charge percentage predicted at 130 minutes; retain the sign

The observed time range is 10–50 minutes inclusive. How is a prediction at 50 minutes classified?

What follows from an input lying inside the observed range?

The battery equation predicts −2% outside its fitted range. What should be reported?