One-sided test

A one-sided test states $H_a$ with a strict inequality, $<$ or $>$, taken from the research question before any data are collected. The p-value is then the area in the single tail $H_a$ points to. The null is still tested at the boundary of equality, so $H_0$ carries an equals sign whichever way the alternative faces, and the null distribution is built from that boundary value.

Direction is not a detail of the arithmetic; it decides which area you read. Take $z = 1.75$ from a test of $H_0: p = 0.50$. Against $H_a: p > 0.50$ the p-value is $P(Z > 1.75) = 0.0401$ and you reject at $\alpha = 0.05$ (alpha). Against $H_a: p < 0.50$, on exactly the same data, the p-value is $P(Z < 1.75) = 0.9599$ and you are nowhere near rejecting.

The error that matters here is "the data came out high, so use the upper tail." If you always pick the tail your data fell in, you reject whenever $|z| \ge 1.645$, and under a true null that happens with probability 0.10. You advertised a 5 percent false-alarm rate and delivered 10 percent. A one-sided test is honest only when the direction was fixed before the data existed, and the price of choosing afterwards is precisely that doubling.

The trade runs both ways. A one-sided test needs less evidence to detect an effect pointing the way $H_a$ does, and it gives up the other direction completely: if the statistic lands on the far side of the null value from $H_a$, the p-value is above 0.5 however extreme it is, climbing toward 1 as the contrary evidence strengthens. When the question is only whether something changed, the two-sided test is the default. The chi-square tests in Unit 3 have no one-sided version at all, because squaring the differences throws direction away.

The Fall 2026 course sets up tests in topics 3.5 and 4.4, the Setting Up a Test topics for proportions and for means. Choosing the direction of $H_a$ belongs to that step, not to the calculation after it.