IB Maths AI Answer Templates: Interpret, Justify, Conclude

Linear models

• Template (gradient): "For each extra [1 unit of x, with units], [y] increases (or decreases) by [gradient, with units]."
• Template (intercept): "When [x = 0, in context], [y] is [intercept, with units]." Add "(the fixed cost)" or similar if it has a meaning.

Exponential and sinusoidal models

• Exponential y = k × aˣ + c: "k + c is the value when x = 0; a is the growth factor each [unit], so it grows by (a − 1) × 100% each [unit]; c is the long-term value the model approaches."
• Logistic (HL): "L is the carrying capacity: the maximum [population] the model predicts in the long term."
• Sinusoidal y = a sin(b(x − c)) + d: "ǀaǀ is the amplitude (half the difference between the highest and lowest values); d is the principal axis, the mean level; the period is 360°/b: the time for one full cycle."

• Describe r: "r = 0.912 shows a strong positive linear correlation between [x] and [y]." Use strong (ǀrǀ ≥ 0.75), moderate, weak or very weak, and always say linear.
• Reliability of a prediction: "The prediction is reliable/unreliable because x = [value] is inside/outside the range of the data (interpolation/extrapolation) and the correlation is strong/weak." You need both reasons for full marks when the question asks "is this reliable?".
• Correlation vs causation: "A strong correlation doesn't show that [x] causes [y]; both might depend on [a third factor]."

Comparing two data sets

• Template: "[Group A] has a higher median ([value] vs [value]), so A typically [context, e.g. waited longer]. [Group A] has a larger IQR ([value] vs [value]), so A's [context] are more spread out / less consistent."

χ² test for independence

• H₀: "[Variable A] and [variable B] are independent." H₁: "[Variable A] and [variable B] are not independent."
• Evidence: "p-value = 0.0631 > 0.05 (the significance level)" or "χ²_calc = 5.52 < 5.99 (critical value)".
• Conclusion: "There is insufficient evidence at the 5% level to reject H₀, so [favourite subject is independent of year group]." Or: "Reject H₀: there is evidence at the 5% level that A and B are not independent."

χ² goodness of fit

• H₀: "The data fits the [uniform / binomial B(4, 0.5) / normal] model." H₁: "The data does not fit the model."
• Degrees of freedom: number of categories − 1 (HL: subtract one more for each parameter you estimated from the data).

t-test for two means

• H₀: μ_A = μ_B. H₁: μ_A > μ_B (one-tailed) or μ_A ≠ μ_B (two-tailed), where μ is the population mean [score, time…].
• Conclusion: "p = 0.00697 < 0.05, so reject H₀: there is evidence at the 5% level that the mean [score] of A is greater than that of B."
• Assumptions (if asked): "both samples are random and independent, and both populations are normally distributed."

HL tests and intervalsHL ONLY

• Binomial test of a proportion: "X ~ B(20, 0.3) under H₀: p = 0.3, H₁: p > 0.3. P(X ≥ 10) = 0.0480 < 0.05, so reject H₀: there is evidence that more than 30% of seeds fail."
• Confidence interval: "46.1 < μ < 50.3." Interpret: "We are 95% confident that the population mean [quantity] lies between 46.1 and 50.3." If asked about a claimed value: "17 °C is (not) inside the interval, so it is (not) a plausible value for the mean."
• Type I error (in context): "Concluding that [more than 30% of seeds fail] when in fact [30% fail]." Type II: "Not concluding that [more than 30% fail] when in fact [they do]."
• Symbols: write hypotheses with μ, p or ρ (population parameters), not with x̄ or r.

• Nearest site: "Point P lies in the cell of site B, so B is the nearest [fire station]."
• Why a different site might respond: "P is close to the edge between B and C, so the distances are almost equal; [traffic, road layout, availability] could make C quicker."
• Coverage: "The furthest point of the cell from D is the vertex (6, 5), at a distance of 2.83 km < 3 km, so the whole cell is covered." The explicit comparison is the R1 mark.
• Toxic waste dump / new site: "The point furthest from all sites is a vertex of the Voronoi diagram (or on the boundary of the region): [coordinates], which is [distance] from its nearest sites."

• Template: list the TVM entries, then the answer with the right accuracy and a sentence: "N = 60, I% = 6, PV = 20 000, FV = 0, P/Y = C/Y = 12 ⇒ PMT = −386.66, so the monthly payment is $386.66."
• Interest vs rate: "Total interest = total paid − amount borrowed = $3199.36." A rate is a percentage; interest is money.
• Which option is better? Compare the same quantity (total cost, final value) and say which with the numbers: "Option B costs $1240 less in total, so B is better."

• Template: "The model predicts [impossible or unrealistic value] for [x], for example [a negative height / endless growth], so it is only valid for [domain]."
• Context first: reasons must be about the situation in the question. "Fiona can't run for an infinite time, so t is limited by the length of the race" scores; "the model is just a model" doesn't.
• "Explain why the answer is reasonable": compare with the information you were given ("8.9 m is more than the 8.5 m in the question, which is consistent").

• Start from the question's information and finish with the given answer as your final line. Never substitute the given value to check it.
• Show one more significant figure: to show r = 18.5, write r = 18.49… = 18.5.
• Re-enter: if you can't show it, use the given answer in the next part. Examiners note many students give up instead.