← Apply probability distributions
P(X ≥ 20) becomes P(X > 19.5). P(X ≤ 20) becomes P(X < 20.5). P(X = 20) becomes P(19.5 < X < 20.5).
The reason: a count of exactly 20 occupies the whole block from 19.5 to 20.5 on a continuous scale. Skip the correction and your answer is slightly but consistently wrong. ⚠
8 Choosing & justifying a model
Justify the fit
- Binomial: fixed number of independent trials, two outcomes, constant probability.
- Poisson: events occurring randomly at a constant average rate; count in a fixed interval.
- Normal: continuous measurement, roughly symmetric and bell-shaped.
- State the conditions and check they’re reasonable: then comment on how well the model fits (limitations).
⭐ Identify the assumption, then argue whether it holds
- Questions are marked in two halves: “Assumption identified” AND “explanation of why it is invalid in context.” Naming it is only half the marks.
- Independence — one event does not affect another. 2025: “one post being short doesn’t affect the length of another post.”
- Constant rate (Poisson) — 2024: “the Poisson distribution assumes that the rate of belly-button washing per week remains constant.”
- Constant probability (binomial) — p is the same for every trial.
- Then challenge it using the context given: people wash more in summer, so the rate is not constant; posts by the same author are related, so they are not independent.
Saying “it’s random” is not an assumption
Generic answers — “we assume the data is random”, “we assume the model fits” — earn nothing. You must name a specific assumption of the distribution you chose (independence, constant rate, constant probability, fixed number of trials) and then say, in the context of the question, one concrete reason it might fail.Inverse calculations — working backwards
- You are given a probability and asked for a parameter or a cut-off value.
- Normal, find the value: use inverse normal with the given tail area, μ and σ.
- Normal, find μ or σ: convert to z first — z = (x − μ)/σ — then solve for the unknown. Two unknowns need two given probabilities and simultaneous equations.
- Binomial/Poisson: usually trial and improvement on n or λ until the probability matches. Say what you tried.
- Always sketch and shade first so you use the correct tail — the single biggest source of error here.
Continuity correction
When you approximate a discrete variable (a count) with the continuous normal distribution, widen the interval by 0.5 at each end.P(X ≥ 20) becomes P(X > 19.5). P(X ≤ 20) becomes P(X < 20.5). P(X = 20) becomes P(19.5 < X < 20.5).
The reason: a count of exactly 20 occupies the whole block from 19.5 to 20.5 on a continuous scale. Skip the correction and your answer is slightly but consistently wrong. ⚠