Hypothesis tests for a Normal meanEdexcel A-Level Maths: Revision notes

If $X\sim N(\mu,\sigma^2)$ and a random sample of size $n$ is taken, the sample mean has distribution $\bar X\sim N\left(\mu,\frac{\sigma^2}{n}\right).$ Its standard deviation, $\frac{\sigma}{\sqrt n}$, is called the standard error. It is smaller than $\sigma$ because averages vary less than single values, and it decreases as $n$ increases. No proof is needed. Example: $\sigma=4$, $n=25$: $\bar X\sim N\left(\mu,\frac{16}{25}\right)=N(\mu,0.64)$ and the standard error is 0.8.

The test is about the population mean $\mu$. For a claimed value $\mu_0$: $H_0:\mu=\mu_0\qquad H_1:\mu>\mu_0,\;\mu<\mu_0\text{ or }\mu\ne\mu_0.$ Hypotheses are never written in terms of the sample mean $\bar x$. The test assumes that the population is Normal with a known, given or assumed variance $\sigma^2$, and that the sample is random. Under $H_0$, $\bar X\sim N\left(\mu_0,\frac{\sigma^2}{n}\right)$.

Standardise the sample mean: $Z=\frac{\bar X-\mu_0}{\sigma/\sqrt n}\sim N(0,1^2).$ Then use either:

• the $p$-value: probability of a sample mean at least as extreme as the observed one under $H_0$ (compare with the significance level, halved for a two-tailed test); or
• the critical value of $z$: $\pm1.645$ at 5% one-tailed, $\pm1.96$ at 5% two-tailed, $\pm2.326$ at 1% one-tailed. Example: $\sigma=4$, $n=25$, $H_0:\mu=500$, $H_1:\mu<500$, $\bar x=498$. $z=\frac{498-500}{0.8}=-2.5$, $p=0.0062<0.01$, so reject $H_0$ at the 1% level.

You may find the critical region for $\bar X$ directly. Under $H_0$, the critical values are $\mu_0\pm z\frac{\sigma}{\sqrt n}$. Example: $\sigma=1.5$, $n=36$, $H_0:\mu=12$, $H_1:\mu\ne12$, 5% level. The standard error is $0.25$. Critical values $12\pm1.96\times0.25=11.51$ and $12.49$. The critical region is $\bar T<11.51$ or $\bar T>12.49$. A sample mean outside this region means $H_0$ is not rejected. For a one-tailed lower test at 1% with $\sigma=12$, $n=40$: $\bar L<150-2.326\times\frac{12}{\sqrt{40}}=145.6$.

State what happens to $H_0$, then say what that means in the context:

• Reject $H_0$: 'there is evidence at the 5% level that the mean lifetime of the batteries is less than 150 hours.'
• Do not reject $H_0$: 'there is insufficient evidence at the 5% level that the mean has changed.' Do not say the claim is proved true or false. The test also depends on the assumptions: the population is Normal with the stated variance and the sample is random.