In statistics, picking the method is the whole problem.

Two questions decide most problems before any calculation

Intro statistics rewards a habit that has nothing to do with computation: reading the problem for two facts before touching a formula.

1. What kind of data is this, and how many groups? One proportion, one mean, two means, paired means, categorical counts. That single reading eliminates most of the formula sheet.
2. Is the sampling independent? "Without replacement", "the same subjects before and after", "drawn from the same bag" — each of these changes the procedure, and each is usually one clause in a long word problem.

Almost every wrong answer that looks carefully worked comes from getting one of those two wrong and then executing flawlessly.

The confusions that cost the most marks

• Independent vs paired. Before-and-after on the same subjects is paired. Running a two-sample test on paired data throws away the pairing and usually the conclusion with it.
• With vs without replacement. Without replacement, the second draw's probability depends on the first. The words are easy to skim; the arithmetic changes completely.
• P(A and B) vs P(A given B). "Given that" is a conditional and shrinks the sample space. Multiplying unconditional probabilities instead is the single most common probability error.
• Mutually exclusive vs independent. These are not the same thing, and two mutually exclusive events with non-zero probability are never independent.
• σ vs s, and z vs t. Population standard deviation known is rare outside textbook problems. If you estimated it from the sample, it is a t.
• The p-value read backwards. It is the probability of data this extreme if the null is true — not the probability the null is true, and not the probability you are wrong.
• "Accept the null". Failing to reject is not evidence of no effect. Wording alone loses marks on nearly every exam that asks for a conclusion in context.
• Conclusion without context. Most rubrics require the conclusion restated in the problem's own terms. A bare "reject H₀" is a partial answer.

Choosing the procedure

The question asks about…	Usual procedure	Watch for
One proportion	1-prop z-test / interval	np and n(1−p) both ≥ 10
One mean	1-sample t	σ almost never known; check normality or n ≥ 30
Two means, separate groups	2-sample t	Independence; unequal variances is the default
Before and after, same subjects	Paired t on the differences	Test the differences, not the two columns
Counts in categories	Chi-square goodness-of-fit	Expected counts ≥ 5, and it uses counts, never percentages
Two categorical variables	Chi-square independence	Expected = row × column ÷ total
Fixed trials, two outcomes	Binomial	n fixed, p constant, trials independent
Events per interval	Poisson	Mean and variance are both λ
Sample mean's behaviour	Normal via CLT	Standard error is σ/√n — not σ

How to practise so the method sticks

1. Write the procedure name before any numbers. If you cannot name it, the problem is not an arithmetic problem yet.
2. State the conditions and check them explicitly. Most rubrics award marks for this separately from the answer.
3. Sanity-check magnitudes. A probability above 1, a negative variance, or a confidence interval that excludes every observed value means a setup error.
4. Then verify the working — photograph it and get the first wrong line named, including the choice of method.
5. Log which of the two reading questions you got wrong, not which exercise. Misses cluster into "missed the pairing" and "missed the conditional" far more often than into arithmetic.

Being straight with you

This is a coursework and exam-practice tool. It is not a research statistics package, it will not choose an analysis for real data with real consequences, and it should not stand in for a statistician on anything that gets published or submitted to a regulator.

Related: the math checker explains the step-level check, and checking answers without an answer key covers the estimation habits that catch a wrong setup early.