IB Mathematics Applications & Interpretation (AI) HL Internal Assessment Example

The aim is clearly stated and mathematically focused, which gives the exploration a strong direction from the start. This helps the reader understand what the later correlation, regression, and model-comparison work is trying to answer.

The plan is coherent, but the later discussion does not fully follow through on this promise. The student should return to the MSE comparison with more caution, because a slightly smaller MSE alone is not enough to justify selecting a model that gives an unrealistic prediction.

The student signals a sensible progression from observation to analysis. That sequencing works well because it guides the reader step by step rather than jumping straight to formulas without context.

This conclusion is too strong for what the box plots show. A lack of outliers is useful, but it does not guarantee accurate predictions or justify skipping further checks such as residual analysis or model validation.

This is a good local interpretation of the linear model, but the student should connect it more explicitly to the wider investigation. A model example is most useful when it is also compared with the quadratic and cubic predictions so the reader can judge whether the linear estimate is reasonable.

This summary is understandable, but it is still too general for a strong conclusion. The student should synthesize which model is actually preferred, why, and what evidence supports that choice, rather than simply restating that the relationship is complex.

This is a good moment for deeper synthesis. The student should explicitly state what the contradiction means for the overall argument, rather than leaving the issue unresolved after identifying an impossible prediction.

The comparison would be stronger if the student added a short justification beyond MSE, such as whether the cubic prediction remains realistic across the observed domain. That would make the model choice more logically developed and less dependent on one statistic.

The student uses appropriate statistical language here and links the graph to the next step in the analysis. That helps the reader follow the mathematical progression from visual inspection to formal measurement.

The student could strengthen precision by describing the strength numerically as well as verbally. Referring back to $r=0.6171$ and $R^2=0.3808$ would make the interpretation more exact and easier to verify.

The notation becomes inconsistent here: the alternative hypothesis should be written in terms of $\rho$, not $p$. Correcting this would improve mathematical precision and prevent confusion between correlation and p-value.

This is a clear and correct use of statistical notation and terminology. Stating the significance level explicitly improves clarity because it lets the reader evaluate the hypothesis test using a standard benchmark.

The wording is inconsistent with the rest of the report and should be corrected to keep terminology stable. Even small naming errors matter in mathematical communication because they can make graphs look less carefully integrated into the analysis.

The student states the equation accurately, but the surrounding wording could be clearer and more formal. A concise sentence explaining what the coefficient signs imply would communicate the mathematics more effectively than repeating the model name twice.

The mathematical communication becomes unclear here because the cubic equation is written inconsistently, with the constant split off awkwardly. The student should present the full equation in one clean form so the reader can verify the substitution and follow the model without ambiguity.

How should the student present a model that produces a clearly impossible prediction? This is a key communication issue because the equation may be mathematically valid, but the conclusion must also explain why the model is not reliable in practice.

A short table comparing the three models, their equations, and their MSE values side by side would strengthen communication. That would make it easier for the reader to compare the models quickly and see how each representation supports the final decision.

To make the personal approach more impactful, the student could turn this curiosity into a more precise investigative question. A narrower focus—such as whether 100 m performance predicts 200 m performance for elite male U.S. sprinters—would make the investigation feel more independently designed.

This is a strong personal connection because it helps justify why the relationship between the two events matters to the student. Such detail can be used more effectively elsewhere to frame sharper comparisons and more personal interpretations of the results.

This shows a useful attempt to plan the modelling process, but the student accepts a rule too uncritically. The lowest in-sample MSE does not automatically mean the model is truly best; the student should also consider overfitting, realism of predictions, and whether the model behaves sensibly outside the sample.

The student’s question grows naturally from personal sprinting experience, which gives the investigation a genuine purpose. This works well because the topic is not just chosen for convenience; it is linked to lived experience and a clear curiosity that can drive meaningful mathematical exploration.

This is a thoughtful independent choice because the student is not simply using raw data; they are applying a meaningful selection rule to improve comparability. That decision shows initiative and an awareness that external conditions can distort sprint-performance analysis.

How does that personal example change the way the student thinks about removing outliers? This would be a good place to reflect on when an unusual value is genuinely an error versus when it may represent an important real performance difference.

The student reflects on why the mathematics mattered to them personally, which gives the exploration authenticity. This is effective because it shows engagement with learning, not just with finishing a task.

This is descriptive, but it does not yet critically reflect on what the box plot means for the investigation. The student should comment on how the lack of outliers affects the modelling choice, the limitations of the sample, or whether the data still contains other sources of variation.

This is a meaningful interpretive insight because it connects the statistical result to a broader understanding of sprinting. The student is beginning to move from calculation to interpretation, which is central to strong reflection.

The reflection overstates what the hypothesis test can show. A significant correlation does not justify a claim about winning gold medals, so the student should separate statistical association from competitive success and discuss the limits of the evidence more carefully.

This reflection is thoughtful in topic, but it does not explain why the more complex models are actually trustworthy. The student should reflect on whether added flexibility improves understanding or simply fits noise, especially given the unrealistic cubic prediction.

The conclusion would be much stronger if the student explained why a cubic model should be accepted despite the impossible prediction earlier in the report. A reflective conclusion should reconcile contradictions, not simply repeat the chosen model.

This is an important limitation, but the wording is internally unclear: if age, gender, and country were standardized, they are not limitations in the same way as unmeasured variables. The student should reflect more precisely on which factors were controlled and which still remain uncontrolled.

The student uses a relevant course-level statistic appropriately to quantify the strength of the linear fit. This works well because it provides a numerical basis for discussing how much variation the model explains.

The hypothesis test conclusion is correct, but the discussion stops at significance without extending the mathematics. The student should interpret what this means for the strength of evidence and distinguish statistical significance from practical predictive usefulness.

The linear regression equation is applied correctly to make a contextual prediction, which shows good use of mathematics rather than just reporting a formula. That is a solid example of moving from calculation to interpretation.

A residual plot or residual table would make this comparison much more rigorous. That would allow the student to check whether any model leaves systematic error rather than relying only on MSE values.

This prediction is plausible, but the student does not use it to compare the models against each other. The mathematics would be stronger if the predicted values from all three models were juxtaposed so the reader could see why one model is preferable.

This is a serious mathematical issue because the cubic model generates an unrealistic result for an observed data value. The student should not endorse the model as the best fit without addressing why the equation behaves badly and whether coefficient rounding or overfitting has distorted the prediction.

This is the central modelling weakness: the student treats the smallest MSE as sufficient proof of the best model without considering whether the model is sensible or predictive outside the sample. Course-level mathematics should include model validity, not just numerical comparison.

How meaningful is the tiny difference between the quadratic and cubic MSE values? This is worth questioning because a very small numerical improvement may not justify choosing a model that behaves unrealistically for prediction.

There is a factual inconsistency here: the sentence refers to “299m” rather than 200m. This should be corrected because accurate variable naming is essential when using mathematics to make claims about real data.