Hypothesis test for the mean of a Normal distributionAQA A-Level Maths: Revision notes

A hypothesis test decides whether sample data give enough evidence against a claim about a population parameter. The null hypothesis $H_0$ states the claim, such as $H_0:\mu=250$. The alternative hypothesis $H_1$ says how the parameter differs: $\mu<250$ or $\mu>250$ (one-tailed) or $\mu\ne250$ (two-tailed). Hypotheses are about the population mean $\mu$, never the sample mean. The significance level is the probability, assuming $H_0$ is true, below which we reject $H_0$ (commonly $5\%$ or $1\%$). The critical region is the set of sample results that lead to rejecting $H_0$, and the p-value is the probability, assuming $H_0$, of a result at least as extreme as the one observed.

If $X\sim N(\mu,\sigma^2)$ and a random sample of size $n$ is taken, the sample mean has the distribution $\bar X\sim N\left(\mu,\frac{\sigma^2}{n}\right).$ The standard deviation of $\bar X$ is $\frac{\sigma}{\sqrt n}$, called the standard error. It falls as $n$ rises, so sample means vary less than single observations. The test needs $\sigma$ to be known, given or assumed. Example: $\sigma=6$ and $n=36$ give $\bar X\sim N(250,1^2)$ under $H_0:\mu=250$.

1. State $H_0$ and $H_1$ and the significance level.
2. Write the distribution of $\bar X$ assuming $H_0$.
3. Find the $p$-value, or the critical value.
4. Compare: reject $H_0$ if the $p$-value is below the significance level, or if the sample mean lies in the critical region.
5. Write the conclusion in context. Example: $H_0:\mu=48$, $H_1:\mu>48$, $\sigma=5$, $n=40$, $\bar x=49.6$. Under $H_0$, $\bar T\sim N(48,0.7906^2)$, so $z=2.02$ and $p=0.0215<0.05$. Reject $H_0$. The critical value at $5\%$ is $48+1.645\times0.7906=49.30$, and $49.6$ exceeds it, which gives the same result.

If $H_1:\mu\ne\mu_0$, the critical region is in both tails. At the $5\%$ level put $2.5\%$ in each tail, so the critical values of $z$ are $\pm1.96$. Alternatively double the one-tail probability to get the $p$-value and compare it with $5\%$. Example: $\mu_0=30$, $\sigma=4$, $n=16$, so the standard error is $1$. The critical region is $\bar X<28.04$ or $\bar X>31.96$. A sample mean of $31.8$ is not in it. Equivalently, $p=2\times P(Z\ge1.8)=0.0719>0.05$, so $H_0$ is not rejected. At the $10\%$ level, $0.0719<0.10$, so $H_0$ would be rejected.

Always write a two-part conclusion: the decision about $H_0$ and what it means in context. If the $p$-value is less than the significance level, reject $H_0$: 'there is sufficient evidence at the $5\%$ level that the mean journey time is longer than $48$ minutes'. Otherwise do not reject: 'there is insufficient evidence ...'. Never say $H_0$ is proved true or false. The same difference in sample mean gives stronger evidence with a larger $n$, since the standard error is smaller: a mean of $49.6$ gives $p=0.0215$ with $n=40$ but $p=0.156$ with $n=10$. The test assumes the sample is random, the population is Normal, and the standard deviation is known, given or assumed.