Revise Hypothesis tests for Cambridge 9709 (6.5)

Every hypothesis test begins with two competing statements about a population parameter (like the mean, $\mu$, or proportion, $p$). The null hypothesis, $H_0$, represents the status quo or a statement of no effect. It is the default position we assume to be true. The alternative hypothesis, $H_1$, represents the claim we are trying to find evidence for. It's what we will conclude if we find the null hypothesis is unlikely.

Once hypotheses are set, we choose a significance level, $\alpha$. This is our threshold for evidence. A 5% significance level ($\alpha = 0.05$) means we are willing to accept a 5% chance of wrongly rejecting the null hypothesis. We then collect sample data and calculate a test statistic. This standardised value tells us how far our sample result is from the value claimed in $H_0$.

For a sample mean $\bar{x}$ from a Normal distribution with known population variance $\sigma^2$ (or a large sample by CLT):

Test Statistic: $Z = \frac{\bar{X} - \mu_0}{\sigma / \sqrt{n}}$

where $\mu_0$ is the value from $H_0$.

We compare our calculated Z-value to a critical value from the Normal distribution tables, determined by $\alpha$. If the test statistic falls into the critical region (the 'tails' of the distribution), we have significant evidence to reject $H_0$.

Since our conclusion is based on a sample, not the entire population, there's always a chance we've made a mistake. There are two types of errors we can make in hypothesis testing. It's crucial to understand them, especially when considering the consequences of a wrong decision.