Hypothesis tests for a population mean: VCE Specialist Mathematics Units 3 & 4

1. Test a stated population claim

A hypothesis test asks whether observed sample evidence is unusually inconsistent with a specified population claim. For a single population mean, the null hypothesis H₀ gives a reference value μ₀, and the alternative H₁ identifies the departure being investigated. The test does not determine the population mean with certainty. It supplies a decision rule with a stated error risk under its assumptions.

Write hypotheses about μ, not about the observed sample mean x̄. The sample mean is already known once the data are collected; the unknown parameter is what the investigation concerns. A claim that the mean has increased from 50 becomes H₀: μ = 50 against H₁: μ > 50. A decrease uses μ < 50. A change in either direction uses μ ≠ 50.

Choose the alternative from the research question before inspecting the result. Selecting the direction after seeing which way x̄ falls makes unusually favourable evidence easier to obtain than the intended test allows. A two-sided investigation cannot be turned into a one-sided test merely to halve a disappointing p-value.

Define the population and units too. “Mean greater than 50” is incomplete if it is unclear whether μ refers to waiting time, mass, distance or another quantity. The final conclusion must return to that same population, rather than making a claim about every individual observation.

2. Build the null sampling distribution

For an independent random sample from a normal population with known standard deviation σ, assume H₀ temporarily. Then X̄ ∼ N(μ₀, σ²/n), so a standardised test statistic has a standard normal distribution under H₀:

The denominator measures the typical sampling variation of an average, not the spread of individual observations. A deviation of 2 units could be unremarkable when the standard error is 5, but very unusual when it is 0.2. Standardisation measures the discrepancy on the relevant scale.

For a sufficiently large random sample from a nonnormal population, use the normal approximation for X̄. When σ is unknown and a suitable large sample supplies s, use s/√n as an estimated standard error. The resulting z-based inference is approximate. A small unknown-variance sample is not justified by this method merely because a calculator can produce a number.

Check the assumptions before the arithmetic: the sample should represent the intended population, observations should satisfy the independent sampling model, and exact normality or a suitable large-sample approximation must be available. Increasing n reduces sampling variation but cannot remove selection bias or make dependent measurements independent.

3. Translate “as extreme” into the correct tail

The p-value is the probability, calculated under H₀, of a test statistic at least as extreme as the observed statistic in the direction or directions specified by H₁. It measures inconsistency with the null model. It is not the probability that H₀ is true, and it is not the probability that the result happened “by chance” in an undefined sense.

For a two-sided normal test, the two tails beyond ±|z_obs| have equal areas. Double the smaller tail, not any arbitrary cumulative output. For a one-sided test, evidence in the opposite direction produces a large p-value. If H₁ says “greater” but z_obs is negative, do not take its absolute value and manufacture evidence for an increase.

The significance level α is chosen before examining the evidence. Reject H₀ when p ≤ α; otherwise do not reject H₀. A non-rejection is a lack of sufficient evidence for H₁ at that level, not proof that the null value is correct. Use unrounded p-values for a close decision and then report sensible final accuracy.

4. A critical region is the same rule written before data arrive

Instead of computing a p-value after sampling, choose a rejection region whose probability under H₀ is α. For an upper-tail test, reject when Z ≥ z_1−α, where z_1−α = Φ⁻¹(1 − α). For a lower-tail test, reject when Z ≤ z_α. For a two-sided test, reject when |Z| ≥ z_1−α/2.

These regions can be translated back to sample-mean units. An upper-tail critical value is c = μ₀ + z_1−ασ/√n, and rejection occurs for X̄ ≥ c. A lower-tail value uses μ₀ + z_ασ/√n. In a two-sided test, there are two boundary values. A sketch should label the axis as X̄ or Z, according to the scale actually shown.

For continuous normal models, including or excluding a single boundary point does not change the rejection probability. Nonetheless, keep the algebra and stated decision convention consistent. When a rounded observed value lies extremely close to a boundary, use the original precision rather than deciding from a rough diagram.

5. Link tests and intervals only when they match

Away from exact equality at a critical boundary, a two-sided level-α z test rejects μ₀ precisely when μ₀ lies outside the corresponding central 100(1 − α)% confidence interval, using the same standard error and model. This follows by rearranging |x̄ − μ₀|/(σ/√n) > z_1−α/2. The interval collects the null values that would not be rejected by that matching procedure.

The word “matching” matters. A one-sided 5% test does not generally give the same decision as checking a central 95% interval, because the test uses z_0.95 and the interval uses z_0.975. Nor should an interval using one variability estimate be compared as if identical to a test using another. Identify the tail allocation before using interval reasoning as a shortcut.

Statistical significance also differs from practical importance. A very large sample can detect a very small departure from μ₀. Report the estimated difference and its units alongside the test conclusion when context requires judgement. A p-value by itself says nothing about whether the difference is large enough to matter operationally.

6. Use technology to audit the decision

A reliable calculation records the hypotheses, standard error, observed statistic, selected tail, unrounded p-value and comparison with α. Recompute the probability using the sample-mean distribution as a check: for a lower-tail test, Pr(X̄ ≤ x̄ | μ = μ₀) should equal Φ(z_obs). A mismatch often reveals that variance was entered where standard deviation was expected.

A non-routine investigation can hold x̄ − μ₀ and σ fixed while varying n. The statistic's magnitude grows like √n, so the evidence against H₀ becomes stronger. Conversely, a fixed statistically significant departure can become non-significant with a smaller sample. This explains why a decision cannot be inferred from the observed difference alone.

For an algorithm, determine the tail from H₁, calculate the corresponding p, and use a single conditional decision: if p ≤ α, reject; otherwise do not reject. The contextual explanation remains necessary after the numerical branch. The next guide explores the errors this decision rule can make.