Sampling, Estimation and Hypothesis Testing
This domain tests how samples support inferences: sampling methods, standard error, confidence intervals and hypothesis tests. You must choose the correct distribution and state conclusions in the language of the test.
Core rules
- Probability sampling: simple random, systematic, stratified and cluster. Stratified sampling improves precision when strata are homogeneous internally.
- Standard error of the mean = σ/√n; use s/√n when σ is unknown.
- Confidence interval for a mean: x̄ ± z(σ/√n) when σ is known and n is large; use t when σ is unknown and n is small.
- Hypothesis test: state H0 and H1, choose α, compute the test statistic, compare with the critical value, or compare the p-value with α.
- Type I error = rejecting a true H0; Type II error = failing to reject a false H0.
- One-tailed tests place all α in one tail; two-tailed tests split α between tails.
Worked example
A sample of 100 items has mean 52; the population standard deviation is 10. Test H0: μ = 50 against H1: μ ≠ 50 at 5%. Standard error = 10/√100 = 1. z = (52 − 50)/1 = 2. Critical values are ±1.96. Since 2 > 1.96, reject H0. A 95% confidence interval is 52 ± 1.96(1) = (50.04, 53.96), which excludes 50, confirming the test. The p-value is about 0.046, just below 0.05.
Traps
- Using z when σ is unknown and n is small instead of t.
- Using the one-tailed critical value for a two-tailed test.
- Saying a p-value is the probability that H0 is true.
- Saying you accept H0 rather than fail to reject it.
- Confusing standard deviation with standard error.