AP Statistics — probability and sampling distributions

Units 4 and 5 are where AP Statistics stops describing data and starts modelling it, and they are where most of the course's arithmetic lives. Ten questions here, one per idea. The probability half opens with the addition rule asked so that the overlap is the whole point — ten per cent of an invented town owns both a bicycle and a scooter, and those residents have already been counted twice. Independence follows, not as a formula to recall but as a judgement to make: half a school studies a language and half the orchestra does too, so orchestra membership tells you nothing, and the wrong answers are the three ways that comparison is usually botched. Then the multiplication rule without replacement, where the second draw is conditional on the first and treating the two as independent gives a visibly different number.

Random variables come next. Expected value is asked with a distribution whose mean, mode and unweighted average are three different numbers, so that a candidate who ignores the probabilities, or who names the commonest outcome, or who insists the answer must be a score the spinner can actually award, each lands somewhere distinct. Combining variables turns on the rule that catches everyone: variances add for independent variables and standard deviations do not, so three and four give five rather than seven. The binomial setting is tested by elimination — a varying number of trials, a probability that shifts as cards leave the pack, and a measurement rather than a success or failure — and the geometric setting by counting how many failures must come before the first success.

The last three build to the sampling distribution. A z-score is used for what it is for, comparing two scores from distributions with different centres and different spreads. The central limit theorem is asked against a strongly skewed population, because the point of the theorem is the non-normal case, and because the three commonest misreadings — that the population becomes normal, that one sample becomes normal, that the theorem needs normality to start with — are all worth meeting once. The closing question asks what quadrupling a sample does to the variability of its mean: it halves it, since the square root of four is two, and four times the data buys twice the precision rather than four times.

Every town, factory, school and test in the bank is invented, and the questions say so. Nothing is drawn from any College Board publication, released examination, course and exam description, scoring guideline, formula sheet or normal table. There are no graphs or tables — every distribution is described in words, which keeps the questions readable anywhere, though it also means the display-reading that some real Unit 4 and 5 questions require is not practised here.

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• Apply the addition rule to overlapping events, subtracting the overlap exactly once
• Decide whether two characteristics are independent by comparing a conditional proportion with its unconditional counterpart
• Compute a probability for two draws without replacement, adjusting the second for the first
• Calculate an expected value and read it as a long-run average that need not be an attainable outcome
• Combine independent random variables by adding variances rather than standard deviations
• Test a situation against all four conditions of the binomial setting
• Tell the geometric setting from the binomial by what is fixed and what varies
• Use a z-score to compare values drawn from distributions with different means and spreads
• State what the central limit theorem describes, and what it does not say about the population or about one sample
• Work out how the variability of a sample mean changes with the sample size, through the square root

In an invented school of four hundred pupils, half study a foreign language and a quarter play in the orchestra. Among the orchestra members, half study a foreign language. Are the two characteristics independent, and how can that be told?