Language of hypothesis testingEdexcel A-Level Maths: Revision notes

A hypothesis test uses data from a sample to decide whether a claim about a population is believable.

• The null hypothesis, $H_0$, is the claim assumed true at the start, written as $H_0:p=p_0$.
• The alternative hypothesis, $H_1$, is what we suspect instead: $p>p_0$, $p<p_0$ or $p\ne p_0$. Hypotheses are always written in terms of the population parameter $p$ (the probability of success, or proportion in the population), never in terms of the sample statistic $X$. Example: a coin is suspected of being biased towards heads: $H_0:p=0.5$, $H_1:p>0.5$.

The test statistic is the value calculated from the sample, here the number of successes $X$. If $H_0$ is true, $X\sim B(n,p_0)$, and we ask how likely a result like the one observed would be. The significance level is chosen in advance, commonly 5% or 1%. It is the probability of incorrectly rejecting $H_0$ when $H_0$ is actually true. A smaller significance level demands stronger evidence before $H_0$ is rejected.

• A one-tailed test has $H_1$ in one direction ($p>p_0$ or $p<p_0$): the whole significance level goes in that tail.
• A two-tailed test has $H_1:p\ne p_0$: the significance level is split equally between the two tails. A 10% test has 5% in each tail. To decide which tail a result lies in, compare it with the expected value $np_0$ under $H_0$. For $B(30,\frac16)$ the mean is 5, so an observation of 1 lies in the lower tail and 9 in the upper tail.

The critical region is the set of values of the test statistic that lead to rejecting $H_0$. Its boundary is the critical value. The acceptance region is the remaining values. To find an upper-tail critical region, find the smallest $c$ with $P(X\ge c)\le$ the significance level. For a lower tail, find the largest $c$ with $P(X\le c)\le$ the significance level. The actual significance level is the probability of the whole critical region under $H_0$ and is usually a little less than the stated level. For $X\sim B(20,0.5)$ and critical region $X\ge15$ it is $0.0207$.

The $p$-value is the probability, assuming $H_0$ is true, of obtaining a result at least as extreme as the one observed. For $H_1:p>p_0$ and observed value $x$, the $p$-value is $P(X\ge x)$.

• If the $p$-value is less than or equal to the significance level (halved in a two-tailed test), reject $H_0$.
• Otherwise there is insufficient evidence to reject $H_0$. Always conclude in context and with cautious language: 'there is evidence at the 5% level that the proportion of eggs containing a toy is greater than 20%'. A test never proves $H_0$ or $H_1$ true.