OCR section 2.04 gives you two models to choose between: the binomial B(n, p) for counting successes, and the Normal N(μ, σ²) for continuous measurements. You need to select the right one, use it, and know its limitations.
Every probability below is checked against the standard tables. Work through each screen, answer the questions as you go and collect ⭐ stars. Press Start when you're ready.
X ~ B(n, p) counts the number of successes in n trials. Four conditions must all hold:
E(X) = np = 20 × 0.3 = 6.
Var(X) = np(1 − p) = 20 × 0.3 × 0.7 = 4.2.
Standard deviation = √4.2 = 2.05 (3 s.f.).
Where the model breaks: drawing counters without replacement breaks independence and makes p change โ so it is not binomial. Sampling from a very large population is close enough to be modelled as binomial anyway.
X ~ N(μ, σ²) models a continuous variable that clusters symmetrically about a mean โ heights, masses, measurement errors.
Careful with the notation: N(μ, σ²) quotes the variance, not the standard deviation. N(50, 64) means σ = 8, not 64.
Standardise: z = (62 − 50) / 8 = 12/8 = 1.5.
So P(X > 62) = P(Z > 1.5) = 1 − Φ(1.5).
From tables, Φ(1.5) = 0.93319.
∴ P(X > 62) = 1 − 0.93319 = 0.0668 (3 s.f.).
Sanity check: 62 is 1.5 standard deviations above the mean, so only a small tail should be beyond it. About 6.7% is entirely reasonable. ✓
The inverse Normal runs it backwards: given a probability, find the value.
From the inverse Normal, the z-value with 0.9 below it is z = 1.2816.
Un-standardise: a = μ + zσ = 50 + 1.2816(8) = 50 + 10.25 = 60.3 (3 s.f.).
Always sketch the bell curve and shade. It is the only reliable way to know whether you want Φ(z), 1 − Φ(z), or a difference of two.
When n is large, a binomial distribution starts to look like a Normal one โ the bar chart of B(n, p) becomes an unmistakable bell.
Y ~ B(100, 0.4) with n = 100, which is large.
Mean = np = 100 × 0.4 = 40.
Variance = np(1 − p) = 100 × 0.4 × 0.6 = 24.
∴ Y is approximately N(40, 24), so the standard deviation is √24 = 4.90 (3 s.f.).
An important OCR spec detail. The specification requires you to understand that a binomial with large n can be approximated by a Normal distribution โ but it states that questions explicitly requiring calculations using the Normal approximation to the binomial are excluded. So learn the idea and the parameters; you will not be asked to grind out a probability with it.
Tap a scenario, then tap the model that fits best.
Tap a description on the left, then the correct expression.
Binomial: fixed n, two outcomes, independent trials, constant p
Binomial formulae: P(X = r) = nCrpr(1−p)n−r; E(X) = np; Var(X) = np(1 − p)
Normal: symmetric bell; points of inflection at μ ± σ; about 95% within 2σ
Notation: N(μ, σ²) quotes the variance โ N(50, 64) means σ = 8
Standardise: z = (x − μ)/σ; then use Φ(z) โ always sketch and shade
Inverse Normal: given the probability, find z, then x = μ + zσ
Approximation: B(n, p) ≈ N(np, np(1−p)) for large n โ but OCR excludes calculations with it
That is OCR 2.04 โ and these two models are what hypothesis testing tests. Press Finish to see your score.
You've worked through Statistical distributions for OCR A-level Mathematics A. 🎉
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