The hard problems

The ordinary exams ask what a procedure is. A hard DSE question asks you to run it on a small, odd case, and the odd case is the point: it is chosen so that the general result would be a trap. Four habits cover most of them.

1. Write the whole distribution down. When the sample space is small, list it. Three balls, two draws, three equally likely outcomes: you need no theory, only a table.
2. Simplify before you compute. A variable that takes the values ±1 is its own reciprocal. A sum of identical terms is twice one of them.
3. Ask which distribution you are under. Size is computed under H₀, β and power under H₁, a standard error under the null for a test and with estimates for an interval.
4. Read each option as a claim and test it. Several options are true statements about the wrong thing. Take each to an extreme case or a specific number.

Every worked question below is a real DSE paper question.

Four ideas from Lesson 1 account for every question in this set: a table’s cells add to 1, marginals come from adding along an edge, E(X) and E(Y) are linear in the cells, and covariance is E[XY] − E[X]E[Y]. The unknown-entries questions (DSE 2011, 2017) are solved in the lesson itself. Two others need a different idea.

Two questions test whether you know what the sample proportion and mean actually are. Both are covered in Lesson 3, and the answers repay a second look.

Six questions in this set are conceptual statements about estimates and intervals, and all but one are decided by reading each claim literally. The techniques are in Lessons 5 and 6, and the pattern is short.

• A confidence interval is a statement about the procedure. Any option saying the interval contains β “with probability 0.95” is false (DSE 2010, 2016). The option that says “in repeated sampling 95% of intervals contain the true value” is true.
• To shorten an interval, shrink the standard error or the confidence level, as a multiplier. Compare the factors: s ÷ 3 beats n × 4 (a factor of ½) beats a drop in confidence from 90% to 80% (a factor of 0.78) (DSE 2008).
• Check an interval by arithmetic. σ = 2, n = 25, mean 10: the 95% interval is 9.22 to 10.78, so 9.3, 9.8 and 10.6 all lie inside (DSE 2012).
• The interval for a variance divides (n − 1)S² by chi-square points, the larger one first for the lower limit (DSE 2012). Options with n in place of n − 1, or with the points swapped, fail.
• One Bernoulli draw gives a maximum likelihood estimate of 0 or 1, never strictly between (DSE 2019).

The test of one observation (DSE 2012) is the template: size under H₀ fixes the cut-off, β is the other density over the acceptance region. Two others in this set are shorter.

• A diagnostic test for a rare disease, with the null that the patient is healthy. The mistake to avoid, calling a sick patient healthy, is failing to reject a false null: a Type II error, a false negative (DSE 2011).

Four questions ask you to compute a statistic and compare it with a critical value, and Lessons 8 and 9 solve three of them. The fourth has a feature worth seeing: four of its options are statements, and one of them must be picked out by reasoning rather than computing every probability.

The two-class comparison (z = −2.49: significant at 5%, not at 1%), the two district polls (not different, yet one candidate is preferred in one district) and the claimed income with a known variance (z = −1.36, not rejected) are worked in Lesson 9 and Lesson 8.

Two questions from 2009 ask the same thing in two ways: given the four standard assumptions on the errors of a simple regression, which ones do you need for a particular property? The skill is to tie each property to the argument that proves it, which Lesson 12 gives.

In the exam the hardest step is the first: seeing which of the techniques above the question is inviting. Six situations; pick the opening move.

Every DSE inference question above the easy level, in the order of this lesson. Within each set, moderate questions come first, then hard ones. The easy DSE questions are in the lessons for their topic.