Introduction
Statistical inference uses sample data (and the sampling-distribution theory of Chapter 6) to draw formal conclusions about an unknown population — the two main tools, estimation and hypothesis testing, are two sides of the same inferential coin, and this chapter is arguably the practical core of the entire subject.
1. Point and interval estimation
A point estimate is a single numerical guess for an unknown population parameter (e.g., the sample mean, x̄, as an estimate of the population mean, μ). An interval estimate (developed as a confidence interval in point 3) instead gives a range of plausible values, additionally conveying the estimate’s precision or uncertainty — a point estimate alone gives no indication of how much confidence to place in it, which is why interval estimation is generally preferred whenever it is available.
2. Properties of estimators
A good estimator should be unbiased (its expected value equals the true parameter value, so it does not systematically over- or under-estimate on average) and have low variance (individual estimates cluster tightly around the true value rather than varying wildly from sample to sample) — an estimator combining both properties is described as efficient. Occasionally a slightly biased estimator with much lower variance is preferred in practice over an unbiased one with high variance, since the overall typical error can be smaller despite the bias.
3. Confidence intervals
A confidence interval provides a range, calculated from sample data, expected to contain the true population parameter with a stated confidence level (commonly 95%). Critically, the correct interpretation is procedural, not about any single interval: a 95% confidence level means that if the same sampling and interval-construction procedure were repeated many times, approximately 95% of the resulting intervals would contain the true parameter — it is not correct to say there is a 95% probability the true (fixed) parameter lies within one specific, already-calculated interval.
4. Null and alternative hypotheses
Hypothesis testing compares sample evidence against a specific claim about the population. The null hypothesis (H₀) represents the default assumption (typically “no effect” or “no difference”), while the alternative hypothesis (H₁) represents the claim the researcher is actually interested in providing evidence for. A hypothesis test never directly “proves” the null hypothesis true — it can only fail to reject it (insufficient evidence against it) or reject it (sufficient evidence against it, in favour of the alternative) — this careful, asymmetric wording matters for correct interpretation and is frequently tested directly.
5. Type I and Type II errors, significance level, and power
A Type I error is incorrectly rejecting a true null hypothesis (a “false positive”); its probability is controlled by the chosen significance level (α), typically set at 0.05. A Type II error is incorrectly failing to reject a false null hypothesis (a “false negative”); its probability (β) is related to the test’s power (1 − β, the probability of correctly rejecting a false null hypothesis). These two error types trade off against each other for a fixed sample size — reducing the chance of one type of error (e.g., by lowering α) generally increases the chance of the other, and the only way to reduce both simultaneously is to increase the sample size.
6. Tests concerning means, proportions, variances, and other parameters
Which specific test procedure to apply depends on the parameter of interest and the underlying conditions: z-tests (used when population variance is known or sample size is large), t-tests (used when population variance is unknown and estimated from the sample, particularly important for smaller samples), chi-square tests (for variances, or for the categorical applications of Chapter 5), and F-tests (for comparing two variances, and underlying ANOVA in Chapter 8) — correctly matching the test procedure to the specific parameter and known/unknown conditions is the essential applied skill of this entire chapter, and follows a consistent, repeatable sequence: state hypotheses, check assumptions, compute the test statistic, compare to a reference distribution (or p-value) to reach a decision, and finally state a conclusion in the specific context of the original question.
Key Takeaways
- Point estimates give a single value; interval estimates (confidence intervals) additionally convey precision, and are generally preferable when available.
- A good estimator is unbiased (correct on average) and has low variance (consistent across samples); these two properties together define efficiency.
- A confidence interval’s correct interpretation is about the long-run reliability of the procedure across repeated sampling, not a probability statement about one already-calculated interval.
- Hypothesis testing never proves the null hypothesis; it can only fail to reject it or reject it in favour of the alternative, based on the strength of sample evidence.
- Type I error (false positive, rate α) and Type II error (false negative, rate β, related to power) trade off against each other for fixed sample size; only increasing sample size reduces both simultaneously.
- Choosing between z, t, chi-square, and F-type procedures depends on the parameter of interest and whether relevant quantities (like variance) are known or must be estimated.