Inferential statistics depends on how data are collected. In simple random sampling, every member of the population has an equal chance of selection; in stratified sampling, the population is divided into strata and a random sample is drawn from each, which can improve precision when strata differ. Cluster sampling selects whole groups rather than individuals and reduces cost, while systematic sampling selects every k-th element after a random start. Biases threaten validity at every stage: selection bias arises from how units are included, response bias from inaccurate answers, non-response bias from systematic differences between respondents and non-respondents, and sampling bias generally from any non-random mechanism that distorts estimates.
A sampling distribution is the probability distribution of a statistic across many samples of the same size, and its standard deviation is the standard error. For the sample mean, the standard error is \(\sigma/\sqrt{n}\). A point estimate is a single value used to approximate a parameter, while an interval estimate such as a confidence interval gives a range of plausible values. A 95% confidence interval should be interpreted carefully: if sampling were repeated many times, about 95% of such intervals would contain the true parameter. Higher confidence levels produce wider intervals because more certainty requires more room. The sample mean is an unbiased estimator of the population mean, meaning its expected value equals \(\mu\); dividing the sum of squared deviations by \(n-1\) rather than \(n\) in the sample variance — Bessel's correction — likewise makes the sample variance an unbiased estimator of \(\sigma^2\). The maximum likelihood estimator (MLE) is the parameter value that maximizes the probability of the observed data.
Hypothesis testing is the formal procedure for evaluating claims about a population. The null hypothesis \(H_0\) is the default assumption, usually of no effect or no difference; the alternative \(H_1\) is the claim being tested. The p-value is the probability, assuming \(H_0\) is true, of observing data at least as extreme as those obtained. The significance level \(\alpha\) (often 0.05) is a pre-chosen threshold: if the p-value falls below it, the null hypothesis is rejected in favor of the alternative. Two kinds of errors are possible: a Type I error rejects a true null (a false positive), while a Type II error fails to reject a false null (a false negative). Statistical power, \(1 - \beta\), is the probability of correctly rejecting a false null, and it increases with larger samples, larger effects, lower variance, and a higher \(\alpha\). Tests may be one-tailed (a directional alternative such as \(\mu > 5\)) or two-tailed (any difference such as \(\mu \ne 5\)). Even when an effect is statistically significant, it may lack practical significance — meaning it is not large enough to matter in the real world, regardless of p-value.