Plan a Mathematics AI regression investigation

Ask what the model should help you understand

Begin with a relationship and a purpose. An illustrative question is whether journey distance can help predict travel time for a clearly defined set of bus journeys. Define the population, route conditions and time period before looking for a high correlation. Decide what you need the model to do: describe an observed pattern, compare conditions or make a limited prediction.

Keep the question narrow enough that different rows describe comparable observations. Combining unrelated transport modes or different definitions of travel time may produce a tidy graph with an unclear interpretation.

Audit the dataset before fitting anything

Record the source, collection date, variable definitions, units and missing values. If you collect observations, agree the method with your teacher and follow school requirements for permissions and privacy. If you use published data, check whether categories or measurement methods changed during the period. Preserve the original dataset separately from your working version.

There is no universal sample size that makes every investigation convincing. Explain why your observations are sufficient for the question and what coverage is missing. Several measurements of the same journey may also share conditions and should not automatically be treated as fully independent.

Look at the scatter before choosing the model

Plot the observations with labelled axes. Examine the shape, spread, clusters and unusual values. Write down a provisional reason for choosing a linear or another suitable relationship. Investigate an unusual point using the original record. A transcription error can be corrected transparently; a genuine unusual journey may reveal something important about the model.

Choose mathematical techniques you can explain and that your teacher considers appropriate to your course. A fitted equation is useful only when its variables and coefficients have a clear meaning in the investigation.

Use residuals to challenge the fit

For an illustrative observation, suppose the fitted model predicts 18 minutes and the observed time is 23 minutes. The residual, observed minus predicted, is +5 minutes: the model underestimates that journey. Calculate residuals consistently and plot them against the explanatory variable or fitted values.

A curved pattern can suggest that a straight line misses part of the relationship. A widening spread can suggest that prediction becomes less precise for longer journeys. Describe the actual pattern in your plot before proposing a change. Avoid selecting models solely because one summary statistic improves.

Make a limited, testable interpretation

Explain the slope in the correct units and the intercept in context. An intercept outside a sensible range may be a mathematical feature rather than a realistic zero-distance journey. Distinguish an association from a causal explanation: traffic, stops and time of day could influence journey duration alongside distance.

• Keep predictions within the range you investigated unless you justify extrapolation.
• Where feasible, test a prediction using observations not used to fit the model.
• Compare errors with what would be useful for the real decision.
• Explain limitations using evidence from the dataset, rather than a generic list.

Keep an understandable calculation trail

Save the software settings, fitted equation and plots, and show a representative calculation in your own working. Explain any exclusions or transformations. Finish by answering the original question at the strength your evidence supports. If the model is unreliable under certain conditions, that finding can guide a sensible refinement or a more cautious conclusion.