IB Mathematics Analysis and Approaches (AA) HL Internal Assessment Example

The exploration opens with a clear introduction and a focused research direction, so the reader immediately understands the purpose of the investigation. This contributes to the coherent overall structure of the report.

The assumptions section is helpful because it makes the model boundaries explicit before any calculations begin. That improves logical development by showing which real-world factors are being intentionally simplified.

Would the logic be clearer if the student defined the compensation cost as a piecewise function before substituting numbers? A more explicit structure would prevent later confusion in the calculations.

This table is useful, but the information is dense and difficult to scan quickly. The student could improve conciseness by summarizing the full range and highlighting only the values near the optimum, rather than listing every case in the main body.

The discussion repeats the same idea that compensation is zero while profits rise before the threshold. The student should shorten this explanation and focus on the key turning point, which would make the argument more concise without losing meaning.

The paragraph would be stronger if it briefly explained why the plotted trend peaks where it does, rather than restating the figure. Adding one sentence linking the maximum to the balance between extra sales and compensation would sharpen the logical flow.

The heading for this section is broad, but the analysis underneath moves between several scenarios quickly. The student could improve organization by giving each sensitivity test its own short sub-section or labeled paragraph.

The conclusion and evaluation are clearly separated from the body of the investigation, which gives the report a logical ending. This structure helps the reader move from results to reflection in a controlled way.

The variable definition is not mathematically precise: “additional tickets oversold that is more than 555” is ambiguous and grammatically unclear. The student should define variables in a way that specifies exactly what is being counted and whether it is a random variable or a fixed parameter.

The notation $E(X)=np$ is correctly associated with a binomial model, but the student should state what $n$ is before using the formula. Without that link, the reader cannot verify whether the expected value is being applied to show-ups or to excess passengers.

The student uses standard mathematical language effectively here by introducing profit as revenue minus cost. That concise framing makes the later notation easier to follow.

The expression for expected compensation is not communicated clearly as written. The student should use notation that makes the count of denied passengers explicit, because the current form obscures the mathematical meaning of the calculation.

The figure caption could be more precise about what the table actually contains. Naming the key variables in the caption would help the reader connect the table to the formulas more efficiently.

The transition from the table to the plotted profit values shows good communication structure: numerical results are first established, then visualized. This is an effective way to support the reader’s understanding.

The graph element is clearly described with its axes and range, which supports interpretation of the plotted relationship. That is a good example of using a visual representation to communicate mathematical results.

A legend or direct labeling for each series would make the comparison easier to read at a glance. This would improve the effectiveness of the multiple representations without changing the mathematics itself.

The choice of a real airline overbooking problem is a strong sign of initiative because the student selected a context with practical relevance rather than a purely textbook scenario. That helps the work feel authentic and purposeful.

The student shows a clear personal motivation by linking the investigation to an interest in finance. That personal connection gives the exploration purpose and supports independent decision-making.

To push the investigation further, the student could justify why expected profit is the most appropriate objective in this context, rather than a risk-adjusted measure. That would show more independent thinking about decision criteria.

The sensitivity analysis is useful, but it still relies mainly on parameter substitution. A more original approach would compare results under different show-up probability estimates or a different risk metric, which would better demonstrate creativity.

Changing compensation cost to test how the optimum shifts shows good initiative. The student is not only solving the base case, but exploring how the model behaves under altered conditions.

How would the recommendation change if the airline cared more about avoiding denied boarding than maximizing mean profit? Considering a second objective would show stronger personal judgment.

The definition of $X$ is still inconsistent with the model. Writing $X\sim B(s,p)$ only makes sense if $X$ counts show-ups among the oversold tickets, not the “excess over 555.” The student should redefine the random variable so the binomial model and the verbal description match exactly.

The profit model is a sensible core choice for this investigation: linking revenue from oversold seats to compensation cost makes the mathematics directly relevant to the real decision being studied. This gives the exploration a clear purpose and supports later sensitivity analysis.

This calculation mixes the expectation of total show-ups with excess over capacity, which causes the negative values later in the table. A cleaner approach would define denied boardings as $\max (0,X-555)$ or a piecewise expression before substituting into the compensation term.

The statement that $E(X)$ is the “expected number of total passengers showing up” is inaccurate in this context. The student needs to distinguish clearly between total show-ups, excess passengers above capacity, and denied boarding, because these are different quantities and lead to different formulas.

Consider replacing the integer trial-and-error table with a fully defined expected-profit function, then justify the optimum analytically or with a calculator graph. That would strengthen the mathematical method and make the optimization step more rigorous.

The compensation expression is not correctly written: multiplying by $555$ here does not represent the number of bumped passengers. The student should express expected compensation in terms of the expected denied boardings, not the seat capacity itself.

What changes if the student compares the full distribution of denied boardings rather than only the expected value? Thinking about tail risk would make the profit recommendation more realistic.

The identification of $s=29$ as the profit maximum is clearly stated and tied back to the context. This shows the student can interpret the numerical results and convert them into a meaningful business conclusion.

The sensitivity check is a strong extension because it shows the student can test how the optimal oversell level responds to altered assumptions. This helps demonstrate that the model is being used as a decision tool rather than just a one-off calculation.

To make the sensitivity analysis more persuasive, the student could report the actual change in maximum profit alongside the new optimum. That would show not only where the peak occurs, but how strongly each assumption affects the outcome.

A useful next step would be to compare the current model with a normal approximation or a binomial tail-probability calculation. That would let the student discuss both average profit and the probability of serious overbooking outcomes.

The reflection correctly identifies the limits of using expected value, but it remains general. The student should quantify at least one limitation, such as how variability in show-up rates could change the oversell recommendation or increase the risk of compensation costs.