Hypotheses are written before the data

A test begins with two competing statements about a population parameter:

1. The null hypothesis $H_0$ is the claim being tested, always written with equality: $H_0: p = p_0$. It is the status quo, the manufacturer's specification, the "no change" position.
2. The alternative hypothesis $H_a$ is the claim the investigator is looking for evidence of, written with $<$, $>$, or $\ne$.

A company states that 40% of its customers renew their subscriptions. A manager suspects the true rate is lower and surveys 250 randomly chosen customers, of whom 88 renewed. The hypotheses are

$$H_0: p = 0.40 \qquad \text{versus} \qquad H_a: p < 0.40,$$

where $p$ is the proportion of all the company's customers who renew.

Two things never appear in a hypothesis. A $\hat{p}$: the sample proportion will be known exactly once the data arrive, so there is nothing to hypothesize about it, and $H_0: \hat{p} = 0.40$ is a claim about a number that is simply 0.352. And the sampled group: these are claims about all customers, not about the 250 surveyed.

Which form $H_a$ takes is decided by what is being investigated, before any data are seen:

1. "Is the rate lower than claimed?" gives $H_a: p < p_0$.
2. "Is the rate higher?" gives $H_a: p > p_0$.
3. "Has the rate changed, in either direction?" gives $H_a: p \ne p_0$.

Choosing the direction after seeing which way the sample came out is not allowed, and it is not a technicality. A one-sided test puts the whole significance level in one tail, so picking that tail once the data have pointed at it makes results look more surprising than they are. If the sample lands opposite the stated alternative, that is a legitimate outcome: the evidence simply does not support the claim being investigated.

When the question names no direction, or when a change either way would matter, the two-sided alternative is the correct choice, and its p-value will be twice the one-sided value for the same data.

The same three conditions appear, with one substitution that separates a test from an interval:

1. Random: a random sample or randomized experiment. Here, 250 randomly chosen customers.
2. 10%: $n \le 0.10N$, so 250 customers out of a base in the thousands.
3. Large counts: $np_0 \ge 10$ and $n(1 - p_0) \ge 10$. Here $250(0.40) = 100$ and $250(0.60) = 150$.

The counts use $p_0$, not $\hat{p}$, because a test works inside the world where the null is true. The interval in Topic 3.3 used $\hat{p}$ for the opposite reason: no value of $p$ is being assumed there, so the sample's own estimate is the only proportion available. The same substitution governs the standard error, $\sqrt{\frac{p_0(1-p_0)}{n}}$ in a test and $\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}$ in an interval, which is why the two procedures can give slightly different-looking arithmetic on the same data.

As always, the check is the numbers. "Large counts: $250(0.40) = 100$ and $250(0.60) = 150$, both at least 10" is verifiable; "conditions are met" is not.

The significance level $\alpha$ is the threshold for calling evidence strong enough: reject $H_0$ when the p-value is below $\alpha$. Common values are 0.05, 0.01, and 0.10, and the choice belongs at the start, alongside the hypotheses.

Setting it in advance is what keeps the decision rule from being written to fit the result. Choosing $\alpha = 0.10$ after seeing a p-value of 0.06 converts a fixed rule into a description of the outcome. The level is also a genuine choice with consequences: a smaller $\alpha$ demands stronger evidence before rejecting, which makes false alarms rarer and missed real effects more common. Topic 3.8 puts numbers on that trade.

A complete setup, before any computation, therefore states four things: the parameter in context, the two hypotheses, the conditions with their arithmetic, and the significance level. Everything after that is mechanical.