6.4 Sampling and estimation

A population contains all units of interest; a sample is selected to estimate population characteristics. A parameter describes the population, while a statistic is calculated from the sample.

Define the target population before sampling and consider coverage, non-response and selection bias. Larger samples reduce random error but do not automatically remove systematic bias.

A survey of 500 randomly chosen voters is a sample; the proportion supporting a policy in all eligible voters is a population parameter.

A sample statistic is not the parameter itself, and a large biased sample can still mislead.

Simple random, systematic, stratified, quota and cluster samples select units differently. Stratification preserves subgroup representation; cluster sampling selects groups and may increase dependence.

Match the method to the population list and objective, and state how units are chosen. Convenience or voluntary-response samples often under-represent reluctant groups.

To estimate a school’s results across year groups, stratify by year and sample within each stratum rather than sampling only one class.

Random selection within a biased sampling frame cannot represent people missing from the frame.

For independent observations Xᵢ with mean μ and variance σ², the sample mean X̄ has E(X̄)=μ and Var(X̄)=σ²/n. Its standard error is σ/√n (or estimated with s/√n).

Distinguish variability of individual observations from variability of means, and account for finite-population or dependence conditions when relevant.

If σ=12 and n=36, the standard error of X̄ is 2, even though individual values vary with standard deviation 12.

Increasing n reduces standard error by √n, not by n, and does not necessarily reduce measurement bias.

For normal data, X̄ is exactly normal; for large independent samples it may be approximately normal by the central limit effect. Its mean is μ and variance σ²/n.

Standardise with the standard error, state whether σ is known or estimated, and check independence and sample-size assumptions before using normal probabilities.

With μ=100, σ=15 and n=25, X̄~N(100,9), so a one-standard-error interval is 100±3.

The sample mean does not inherit the population variance unchanged; averaging reduces variance by n.

For independent observations with mean μ and variance σ², E(X̄)=μ and Var(X̄)=σ²/n. A linear transformation aX̄+b has mean aμ+b and variance a²σ²/n.

Separate the spread of individual observations from the standard error of the mean, and state independence assumptions before using the formula.

If σ=10 and n=25, the mean has standard deviation 2; quadrupling n halves the standard error.

A sample mean is not guaranteed to equal μ, and multiplying by a changes variance by a², not a.

An estimator T is unbiased for θ when E(T)=θ. Unbiasedness concerns long-run average, not whether one estimate is close to θ.

Check the expectation algebra and compare variance separately; an unbiased estimator can be noisy, while a biased estimator may have lower variance.

The sample mean is unbiased for μ under random sampling, while dividing by n rather than n−1 gives a biased estimate of population variance.

Unbiased does not mean error-free or best for every loss function.

A mean confidence interval is estimate ± critical value×standard error, using a normal or t model as specified. The confidence level describes repeated-method coverage.

Use the correct critical distribution, sample size and known/estimated standard deviation; interpret the result in context of the population mean.

A sample mean 42 with margin 3 gives (39,45) at the stated confidence level.

The fixed population mean is not random after the sample is taken; “95% confidence” is not a 95% probability that this interval contains it.

For a population proportion p, estimate with p̂=x/n and use p̂±critical value√[p̂(1−p̂)/n] when the normal approximation is justified.

Check that expected successes and failures are sufficiently large, keep the interval within [0,1] in interpretation, and state the confidence level.

If p̂=0.40 and n=100, the estimated standard error is √(0.24/100)≈0.049.

A proportion is not a mean of arbitrary measurements, and the normal interval can be poor for small counts or extreme p̂.