One-Sided Test vs Two-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.

A two-sided test states $H_a$ with a not-equal sign, so the p-value collects probability from both tails: it is the chance, under $H_0$, of a statistic at least as far from the null value as yours in either direction. The null still carries an equals sign and the test is still run at that boundary. Use it whenever the question is only whether something changed.

Take $z = 1.75$ from a test of $H_0: p = 0.50$. The upper tail area is $P(Z > 1.75) = 0.0401$, and the two-sided p-value is 0.0801. At $\alpha = 0.05$ (alpha) that fails to reject, while the matching upper-tailed one-sided test gives 0.0401 and rejects. Identical data, opposite verdicts, which is why $H_a$ is written before the data are seen.

Here is the sentence to stop writing: "a two-sided p-value is always twice the one-sided p-value." The doubling needs two conditions, that the null distribution is symmetric about the null value and that the one-sided alternative points the way the data actually fell. Keep $z = 1.75$ and set $H_a: p < 0.50$. That one-sided p-value is 0.9599, and twice it is 1.92, which is not a probability. The safe statement is that the two-sided p-value is twice the tail area beyond the observed statistic on the side where it landed.

Symmetry is a real condition. Test $H_0: p = 0.30$ with 10 successes in 20 trials against the exact binomial null instead of a normal model. The upper tail $P(X \ge 10)$ is 0.0480, so doubling would report 0.0959, but the outcomes at least as far from the null mean of 6 are $X \ge 10$ and $X \le 2$, which total $0.0480 + 0.0355 = 0.0834$. The $z$ and $t$ models this course uses are symmetric, so doubling is safe there; the reason is the model, not the word two-sided.

A two-sided test at $\alpha = 0.05$ usually matches the 95% confidence interval, though for proportions the two can disagree in close cases, because the test builds its standard error from the null value and the interval from the sample.