Hypothesis tests for a mean

A test asks how unusual the observed statistic would be if the null model were true. State H₀, H₁ and significance level first. For a normal population with known variance, the standardised sample mean uses $z=(\bar x-\mu_0)/(\sigma/\sqrt n)$. A p-value measures a tail probability under H₀, not the probability that H₀ is true.

Organise a z-test around the claim

The source’s eight steps are: define X and population mean μ; state the distributions of X and the sample mean under $H_0$ (including CLT justification if needed); state hypotheses; state the test statistic; state α; specify rejection using a p-value or critical value; calculate; conclude in context.

Its scenario assumes normal masses with variance $80\ \mathrm{kg}^2$, samples 100 men and observes mean 77.2 kg. The claim “more than 75 kg” at 5% gives $H_0:\mu=75$, $H_1:\mu>75$ and $\bar X\sim N(75,80/100)$ under $H_0$, with an upper-tail test.

Here $\mu_0=75$ is the hypothesised population mean, not the observed sample mean 77.2; the source paragraph calling μ₀ a sample mean is inaccurate. A test assesses how unusual the observation is under the null model, not the probability that $H_0$ is true. The full mass scenario is worked below; the later examples below apply the large-sample approximation and reverse test decisions to find unknowns.

Why take a sample?

The source contrasts testing a whole population with drawing a practical sample. Its introductory contexts are mean filtration efficiency in product-quality checks and a school mean-height claim of 1.6 metres assessed with a sample of 50 students.

A sample can supply evidence about a population parameter while leaving uncertainty; it cannot prove or disprove a claim with certainty.

The source also lists one-sample, two-sample and paired t-tests, ANOVA and chi-squared tests as examples of other test families. This chapter concentrates on a population mean using a z-test, with the exact or approximate reference model justified for the case.