Hypothesis testing is a disciplined way to ask “is the data weird enough to abandon the default story?” You write the default as $H_0$ and the rival as $H_1$, pretend $H_0$ is True, and compute how unlikely the data would be in that world — the p-value. If it’s below your tolerance $\alpha$ (typically 0.05), you reject $H_0$. The mechanics for testing a mean against $\mu_0$ is just a Z-score and a normal-table lookup.
✓ Intro · expand
Try first (productive failure)
Before the worked example: spend 60 seconds taking your best shot at this.
A guess is fine — being briefly wrong about a problem makes the explanation
land harder when you read it. This appears once per tutorial; skip
if you already know the trick.
60s
✓ Try first · expand
Worked example
A factory claims its widgets weigh 100g on average, with known population standard deviation 5g. You sample 25 widgets and find a sample mean of 102.5g. Test the claim at significance level $\alpha = 0.05$ (two-sided). Reject $H_0$ (1) or fail to reject (0)?
✓ Worked example · expand
Practice 1 of 3Type a fraction, decimal, or expression — mathjs parses it.
✓ Practice · expand
Reflection
Notice the asymmetry: we can <em>reject</em> $H_0$, but we never “accept” it — only fail to reject. Why is that the right verb? And when you compute the p-value, what exact probability statement is it — about the data, about $H_0$, or about something else?