WST03 Statistics 3

Lessons in WST03

1. Linear combinations of random variables
   Means combine with the same signs as the variables; variances of independent variables always add, with the constants squared. Normal in, normal out.
2. Sums of observations: X1 + X2 + ... + Xn versus nX
   Sum of n separate observations: variance nσ². One observation times n: variance n²σ².
3. Comparing normal variables: P(X > Y) and capacity problems
   Move everything to one side: P(X > Y) = P(X - Y > 0). Then combine means and add variances (squared constants).
4. Populations, sampling frames and simple random sampling
   A simple random sample gives every possible sample of size n the same chance. Describe: frame, numbering, random numbers, rejecting invalid and repeated numbers
5. Stratified sampling
   Stratum sample = stratum size ÷ N × n, then simple random sampling within each stratum.
6. Systematic and quota sampling, and choosing a method
   Systematic: k = N/n, random start from 1 to k, every k th. Quota: fill target numbers, not random, no frame needed.
7. Estimators, bias and standard error
   Unbiased means E(T) = θ; bias = E(T) - θ. Of two unbiased estimators prefer the smaller variance. Standard error of X̄ is σ/sqrt(n).
8. Unbiased estimates of the population mean and variance
   x̄ = Σx/n and s² = (Σx² - nx̄²)/(n - 1) are unbiased for μ and σ². Divisor n gives a biased variance.
9. The distribution of the sample mean
   X̄ ~ N(μ, σ²/n) for a normal population. Standardise with σ/sqrt(n), and round sample sizes up.
10. Confidence intervals for a normal mean (known variance)
    x̄ ± zσ/sqrt(n) with z = 1.6449, 1.96, 2.3263 or 2.5758. Wider for more confidence, narrower for larger n.
11. Interpreting confidence intervals and linking them to tests
    About 95% of intervals made this way contain μ. Count intervals with a binomial. A value outside the 95% interval is rejected by a two-tailed 5% test.
12. Hypothesis test for a normal mean (known variance)
    H0 and H1 in terms of μ; z = (x̄ - μ0)/(σ/sqrt(n)); z values to 4 d.p.; conclude in context with direction.
13. The Central Limit Theorem and large samples
    For large n, X̄ ≈ N(μ, σ²/n) for any population; use s for σ when n is large. Say 'n is large, so by the CLT'.
14. Difference between two means (known variances)
    X̄ - Ȳ ~ N(μx - μy, σx²/nx + σy²/ny). Test with z = ((x̄ - ȳ) - d)/standard error; interval (x̄ - ȳ) ± z × standard error.
15. Difference between two means (large samples)
    Large samples: use sx² and sy² for the variances, justified by the CLT and large n. Fix H1 from the wording before calculating.
16. Goodness of fit: the chi-squared statistic
    X² = Σ(O - E)²/E; ν = cells - 1 - parameters estimated; combine cells with E < 5; reject for large X².
17. Goodness of fit: binomial and Poisson models
    Estimate λ by x̄ and p by x̄/n; last cell 'or more'; combine E < 5; subtract 1 extra degree of freedom for each estimated parameter.
18. Goodness of fit: normal and continuous uniform models
    Uniform: E proportional to class width. Normal: E = N × class probability with open end classes. Subtract one degree of freedom per estimated parameter.
19. Contingency tables
    E = row total × column total ÷ grand total; ν = (r - 1)(c - 1); H0 no association; compare O with E to describe any association.
20. Spearman's rank correlation coefficient
    Rank both variables the same way, find Σd², r_s = 1 - 6Σd²/(n(n² - 1)). Spearman's measures agreement of rankings, so it suits ranked or non-linear monotonic
21. Testing Spearman's rank correlation
    H0: ρ_s = 0; one-tailed or two-tailed from the wording; critical value from the Spearman table (halve the level per tail for two-tailed); conclude in context.
22. Testing for zero correlation with the PMCC
    H0: ρ = 0; PMCC table (one-tailed levels); assumption bivariate normal; otherwise use Spearman's.
23. S3 exam technique
    Parameters in hypotheses; z to 4 d.p.; standard errors with sqrt(n); correct degrees of freedom; conclusions in context with direction.

Next exams for WST03

• January 2027: Fri 22 Jan 2027, afternoon, 1h 30m
• Summer 2027: Wed 9 Jun 2027, afternoon, 1h 30m

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