Course 1 · Week 3 — Sampling, estimation, one-sample inference

Cheatsheet — biostats_courses

Author

R. Heller

Populations vs samples

Parameter = truth about the population (\(\mu\), \(\sigma\), \(\pi\)).
Estimator = statistic computed on a sample (\(\bar x\), \(s\), \(\hat p\)).
Bias = \(E[\hat\theta] - \theta\). MSE = bias² + variance.

Central limit theorem

For iid samples with finite variance, as \(n \to \infty\): \(\bar X \sim N\!\left(\mu, \sigma / \sqrt n\right)\)

Holds roughly by \(n \approx 30\) for most non-pathological distributions.

Standard error

\(\mathrm{SE}(\bar X) = \sigma / \sqrt n\), estimated by \(s / \sqrt n\).

Bootstrap

B <- 2000
boot_mean <- replicate(B, mean(sample(x, replace = TRUE)))
quantile(boot_mean, c(0.025, 0.975))   # 95% percentile CI
sd(boot_mean)                           # bootstrap SE

Permutation test

obs  <- mean(a) - mean(b)
pool <- c(a, b)
null <- replicate(5000, {
  idx <- sample(seq_along(pool), length(a))
  mean(pool[idx]) - mean(pool[-idx])
})
mean(abs(null) >= abs(obs))   # two-sided p

Maximum likelihood

Given data \(x\) and model \(f(x; \theta)\):

\(\ell(\theta) = \sum \log f(x_i; \theta)\).
\(\hat\theta_{\text{MLE}} = \arg\max_\theta \ell(\theta)\).
Fisher information \(I(\theta)\); asymptotic SE = \(1/\sqrt{I(\hat\theta)}\).

One-sample tests (five-step template)

Hypothesis → Visualise → Assumptions → Conduct → Conclude.

Test	R	Assumptions
One-sample t	`t.test(x, mu = 0)`	roughly normal or large n
One-proportion	`prop.test(k, n, p = 0.5)` or `binom.test`	\(np, n(1-p) \geq 5\) for `prop.test`

Hypothesis-testing vocabulary

Type I: reject true H₀ (probability \(\alpha\)).
Type II: accept false H₀ (probability \(\beta\)).
Power = \(1 - \beta\).
A p-value is not the probability that H₀ is true.

Decision rule for Week 3

Want a CI? Bootstrap first, then ask whether a formula applies.
Want a test? State H₀ and α in writing before running it.
Want to know if n is big enough? Simulate the CLT for your outcome.

Common pitfalls

Reporting SE when you meant SD (or vice versa).
Using the 95% CI to “prove” there is no effect.
Running many tests and quoting the smallest p-value.