Further Statistics 1

What further statistics 1 covers

One of the eight optional papers of 9FM0, sat as Paper 3B, so revise it only if your centre teaches it. It extends the statistics half of A level Mathematics with new families of distributions and the tests that compare a model with data. It is also the gateway to Further Statistics 2, which may be taken only as a matching pair with it.

The main ideas

• Discrete random variables: expectation, variance, the expected value of a function of X, and the coding results.
• The Poisson distribution, its conditions, its equal mean and variance, the additive property, and its use to approximate a binomial.
• The geometric and negative binomial distributions, for the number of trials up to a first or an rth success.
• Hypothesis tests on a Poisson mean and on a geometric parameter, with critical regions and actual significance levels.
• The Central Limit Theorem, giving a sample mean and a sample total approximate normal distributions once n is large.
• Chi-squared goodness-of-fit tests, with small expected frequencies pooled and a degree of freedom removed for each estimated parameter, then contingency tables.
• Probability generating functions for means, variances and sums of independent variables, and the size and power of a test.

The results it turns on

E(X) = Σx P(X = x), and Var(X) = E(X²) − [E(X)]²

the mean and variance of a discrete distribution

E(aX + b) = aE(X) + b, while Var(aX + b) = a²Var(X)

coding, where the shift drops out of the variance

X ~ Po(λ), with E(X) = Var(X) = λ

the Poisson model and its own suitability check

χ² = Σ(O − E)²/E

the statistic for both goodness of fit and contingency tables

expected = row total × column total ÷ grand total, with (r − 1)(c − 1) degrees of freedom

a contingency table

G(t) = E(t^X), with mean G'(1) and variance G''(1) + G'(1) − [G'(1)]²

probability generating functions

Where it usually goes wrong

• The Poisson parameter has to be scaled to the window the question asks about, since a per-minute rate rarely matches the interval in the stem.
• Degrees of freedom are counted after pooling, and one more is subtracted for each parameter estimated from the data, with that parameter named.
• The second derivative of a generating function at 1 gives the expected value of X(X - 1), which is why the variance formula carries the extra first derivative term.
• Power is one minus the probability of a Type II error, and it needs a specific alternative value before it can be evaluated at all.

Where to start

Discrete random variables first, then the Poisson, then the geometric and negative binomial pair. Hypothesis tests follow once those distributions are secure. The Central Limit Theorem is short and self-contained. Goodness of fit and contingency tables belong together, and generating functions and the quality of tests close the paper.