A portfolio manager claims her strategy beats the S&P 500 by 2% per year. You have 36 monthly returns. Hypothesis testing is the framework that turns that claim into a statistical verdict.
The central limit theorem (CLT) says the sampling distribution of the sample mean approaches a normal distribution as sample size grows, regardless of the underlying population shape, provided the population has finite variance. The accepted threshold is n ≥ 30.
The sample mean has expected value equal to the population mean. Its standard deviation (the standard error) shrinks as n grows.
KEY: The CLT lets you use normal-based inference even when the underlying data are not normal, as long as n is large enough. Returns are skewed and fat-tailed, but the mean of 30+ returns is approximately normal.
Sampling Methodologies. Simple random sampling gives every population member equal selection probability. Stratified random sampling divides the population into subgroups (strata) and samples each, useful when subgroups differ systematically. Cluster sampling samples whole clusters, cheaper but less precise.
Common mistakes
- Confusing Type I and Type II errors. Type I = rejecting a true null (α). Type II = failing to reject a false null (β). Trap: "α is the probability of a Type II error."
- Putting the claim in the null. The null always carries the equality. A claim that "return > 0" goes in , not . Trap: .
- Using a one-tailed critical value for a two-tailed test. At α = 0.05, two-tailed z = 1.96; one-tailed z = 1.645. Trap: comparing |z| = 1.80 to 1.645 in a two-tailed test and rejecting.
Bottom line
- CLT: the sampling distribution of the mean approaches normal as n grows, regardless of population shape. Rule of thumb n ≥ 30
- Standard error of the mean = (known variance) or (unknown); it shrinks as n grows
- Use z when variance is known, t when unknown; the t distribution has fatter tails
- Type I = reject true H₀ (α). Type II = fail to reject false H₀ (β). Power = 1 − β; raising n raises power without inflating α
Exam shortcut
For the two-tailed 95% critical z, memorize 1.96; one-tailed 5% is 1.645; 99% two-tailed is 2.58. For test selection: distribution known and n ≥ 30 → z or t; small n and non-normal → non-parametric. For error-type questions: "Type 1 starts with R" (Reject true null); "Type 2 starts with F" (Fail to reject false null).
The full lesson (about 1,917 words, 13 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.
Learning objectives
- estimation and hypothesis testing
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