The idea

Hypothesis testing for a normal mean

Testing a claim about the mean μ of a normal population from a sample, using that the sample mean is N(μ, σ²/n) — forming the test statistic z = (x̄ − μ₀)/(σ/√n) and comparing it to the critical z-value.

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Statistics · Hypothesis testing

Hypothesis testing for a normal mean

Why it works

The sample mean under the model

A machine should produce rods of mean length

50

cm with

\sigma = 4

cm; your sample of 40 rods averages

51.2

cm. Drift, or chance? A single rod of

51.2

cm would be utterly unremarkable — but a MEAN of 40 rods is far more tightly pinned down. If individual values come from

N(\mu, \sigma^2)

, the mean of a sample of

n

of them is itself normal — and less spread out, because averaging cancels out the extremes:

\bar{X} \sim N\!\left(\mu,\ \frac{\sigma^2}{n}\right), \qquad \text{standard error } \frac{\sigma}{\sqrt{n}}.

The standard deviation of the sample mean,

\dfrac{\sigma}{\sqrt n}

, shrinks as

n

grows — bigger samples give means clustered more tightly around

\mu

. This is what makes a sample mean good evidence about

\mu

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