WST02 Statistics 2

Lessons in WST02

1. The binomial distribution as a model
   B(n, p): fixed n, two outcomes, independent trials, constant p. P(X = r) = C(n, r) p^r (1 - p)ⁿ - r.
2. Cumulative binomial probabilities and tables
   Tables give P(X ≤ x). Turn every other statement into that form, and count failures when p > 0.5.
3. The Poisson distribution as a model
   Po(λ): P(X = r) = e^-λ λ^r / r!. Conditions in context: singly, independently, randomly, constant rate.
4. Poisson probabilities, tables and changing the interval
   Tables give P(X ≤ x). Scale λ with the length of the interval: Po(λ) per unit means Po(tλ) per t units.
5. Mean and variance of the binomial and Poisson distributions
   B(n, p): mean np, variance np(1 - p). Po(λ): mean = variance = λ. Mean close to variance supports Poisson.
6. The Poisson approximation to the binomial
   B(n, p) ≈ Po(np) when n is large and p is small (rough guide: n > 50, p < 0.2). Give both conditions. Count failures if p is near 1.
7. Continuous random variables and probability density functions
   f(x) ≥ 0, total area 1, P(a < X < b) = ∫ f(x) dx from a to b, P(X = a) = 0.
8. The cumulative distribution function
   F(x) = P(X ≤ x) = ∫ f(t) dt up to x. Write F for all x, with the 0 and 1 lines. Piecewise: carry the earlier total forward.
9. From cdf to pdf: f(x) = F'(x)
   f = F'. Probabilities come from F(b) - F(a). Use F(top) = 1 to find constants.
10. Mean and variance of a continuous random variable
    E(X) = ∫ x f(x) dx, Var(X) = ∫ x² f(x) dx - [E(X)]². Use symmetry when you can.
11. Mode, median, quartiles and skewness
    Median: F(m) = 0.5; quartiles: F = 0.25, 0.75; mode: maximum of f. mean < median < mode means negative skew.
12. The continuous uniform distribution
    U[a, b]: f = 1/(b - a), F = (x - a)/(b - a), mean (a + b)/2, variance (b - a)²/12.
13. Modelling with the continuous uniform distribution
    Find the real range first, sketch, then use rectangle areas. For E(X²) use Var(X) + [E(X)]².
14. The normal approximation to the binomial
    B(n, p) ≈ N(np, np(1 - p)) for large n, p near 0.5. Always use a continuity correction and σ = sqrt(variance).
15. The normal approximation to the Poisson, and choosing an approximation
    Po(λ) ≈ N(λ, λ) for large λ, with a continuity correction. Scale λ to the interval before you approximate.
16. Populations, censuses and samples
    Population, census (everyone), sample, sampling unit (one member), sampling frame (list of all units). Census: accurate but slow, costly, impossible if destruct
17. Statistics and sampling distributions
    A statistic uses sample values only. Its sampling distribution lists all its values and their probabilities; count the orders.
18. Hypothesis testing: the ideas
    H0 and H1 in terms of p or λ. One tail for 'increased' or 'decreased', two for 'changed'. Reject H0 if P(result at least as extreme) < α. Conclude in context.
19. Critical regions and actual significance levels
    Critical region: values that reject H0, chosen so the tail probability is as large as possible without exceeding α. Actual significance level = P(in critical re
20. Hypothesis tests for a binomial proportion
    Define p, write H0 and H1, state X ~ B(n, p0), compare the tail probability (or critical region) with α, conclude in context.
21. Two-tailed tests
    Two-tailed: H1 uses ≠, each tail at most α/2, actual significance level = sum of both tails.
22. Hypothesis tests for a Poisson mean
    Hypotheses use λ. Scale λ to the observation period. Conclude about the rate, in context.
23. Hypothesis tests using approximations
    Choose the approximation from the sizes of n, p or λ, apply it under H0 (with a continuity correction for normal), then test as usual.
24. S2 exam technique
    Parameters in hypotheses; probabilities behind every critical region; two-part conclusions about a rate; conditions in context; full cdf; continuity corrections

Next exams for WST02

• January 2027: Mon 18 Jan 2027, afternoon, 1h 30m
• Summer 2027: Wed 2 Jun 2027, afternoon, 1h 30m

From Pearson's final timetables. All exam dates. Always confirm your entries with your exams officer.