Confidence intervals for a population mean: VCE Specialist Mathematics Units 3 & 4

1. Estimate the population, acknowledge the sampling variation

A sample mean x̄ is a point estimate of an unknown population mean μ. It is useful but incomplete: another random sample would generally give a different estimate. A confidence interval surrounds the observed mean with a margin that reflects this sampling variability. The aim is to use a method that captures the fixed population mean at a chosen long-run rate.

The uncertainty concerns the unknown population mean, not the variability of individual observations. A 95% confidence interval for a mean is not an interval containing 95% of individual values. Individual values usually vary much more widely than the sample mean. Before calculating anything, identify the population, the quantity whose mean is being estimated, and the sampling procedure that makes the inference credible.

Start from the sampling distribution. Under iid sampling, X̄ has mean μ and standard deviation σ/√n. If the population is normal with known σ, its distribution is exactly normal. For a suitable large sample from a nonnormal population, the normal approximation can be used. The interval inherits the strengths and limitations of that model.

2. Build the interval from a central normal probability

Choose a confidence level C, written as a proportion. Let z* be the positive standard normal quantile satisfying Φ(z*) = (1 + C)/2. For 95% confidence, 2.5% lies in each tail, so z* = Φ⁻¹(0.975) ≈ 1.95996. For 90% confidence use approximately 1.64485; for 99% confidence use approximately 2.57583.

The probability statement is exact for a normal population with known σ, and approximate when normality of the mean is approximate. Rearranging it places μ between two random endpoints. After observing a sample, those endpoints become numbers. The margin of error is M = z*σ/√n and the interval's total width is W = 2M.

Always distinguish a half-width from a full width. If a question requests an interval no wider than 4 units, the allowed margin is 2 units. If it requests an estimate within 4 units, the intended margin is usually 4 units. Read the wording before solving for a sample size. A factor-of-two mistake here becomes a factor-of-four mistake in n.

3. Interpret coverage without turning μ into a random draw

Imagine collecting many independent samples and calculating one 95% interval from each. About 95% of the resulting intervals would contain μ under the model. The parameter remains fixed while the sample mean and interval endpoints move. A particular interval either contains μ or it does not; we usually cannot tell which because μ is unknown.

It is therefore inappropriate, within this frequentist method, to assign a 95% probability to the fixed μ being inside an already observed numerical interval. Before sampling, it is correct to discuss the probability that the random interval will contain μ. After sampling, use the language of confidence and explain the method's repeated-sampling coverage.

The nominal confidence level does not promise that exactly 95 out of the next 100 intervals will contain μ. The number covering μ is itself random. A simulation returning 943 successful intervals out of 1000 is not automatically a failed method. Conversely, a simulation producing exactly 950 successes is not proof that every assumption has been satisfied.

4. Use a large-sample standard-deviation estimate carefully

Often σ is unknown. For a sufficiently large random sample, replace it by the sample standard deviation s and use the approximate interval x̄ ± z*s/√n. Here s measures spread among observations; s/√n estimates the spread of sample means. The sample standard deviation uses deviations from x̄ and the divisor n − 1. Do not confuse it with a calculator's population-standard-deviation output.

This replacement creates an additional approximation. A small sample from a population with unknown σ is not made suitable for this z-based method merely by inserting s. The Specialist method here is the stated large-sample approximation, rather than a different small-sample inference procedure. Strong skewness and rare extremes still require care even when n passes a commonly used guideline.

No confidence interval formula repairs biased selection. A huge sample drawn only from people who volunteer may be very precise about an unrepresentative group. Independence, the population to which the sample refers, and the suitability of the distributional approximation should accompany the numerical answer.

5. Trade confidence, precision and sample size

At fixed n and σ, higher confidence needs a larger z* and therefore a wider interval. Capturing μ more often requires tolerating a broader range of plausible estimates. At fixed confidence and σ, larger samples make the interval narrower. Increasing σ makes it wider because the original observations are more variable.

For a target margin M₀ with known σ, solve z*σ/√n ≤ M₀. This gives n ≥ (z*σ/M₀)². Round upward to an integer, then verify the achieved margin. When σ is only a planning estimate, the calculation gives a planned sample size, not an unconditional promise about a future interval based on a different realised s.

These relationships are quantitative. Halving the margin requires four times the sample size at the same confidence. Doubling the confidence percentage is not a meaningful way to predict width: confidence cannot exceed 100%, and the quantile changes nonlinearly. Use the quantile rather than treating “95%” as a multiplicative factor.

6. Simulate interval coverage and investigate a reverse problem

A coverage simulation must know the population mean used to generate observations. It then checks whether each newly calculated interval contains that known value. Generate a fresh sample each time and let its mean move. For known σ, the margin remains fixed when n and confidence are fixed; if s is used, the margin varies too.

input n, R, known μ, known σ, confidence C
z ← inverseNormal((1 + C) / 2)
covered ← 0
for r from 1 to R
    sample ← n fresh independent population draws
    mean ← average(sample)
    margin ← z × σ / sqrt(n)
    if mean − margin ≤ μ and μ ≤ mean + margin then
        covered ← covered + 1
    end if
end for
output covered / R

Run the simulation at several confidence levels using the same n. The covering fraction should broadly increase with confidence because each sample's interval becomes wider. When using a nonnormal population, compare different sample sizes to investigate whether nominal and simulated coverage agree. A discrepancy can reflect approximation error, simulation noise or faulty code; inspect all three possibilities.

A useful reverse problem starts from interval endpoints. Their midpoint gives x̄ and half their difference gives M. If σ and n are known, z* = M√n/σ and the implied confidence is C = 2Φ(z*) − 1. For example, a margin of 3 with σ = 10 and n = 100 gives z* = 3 and C ≈ 0.99730. Width alone cannot reveal confidence without the sample size and variability.