A one-tailed (one-sided) test is a significance test whose alternative hypothesis specifies a direction, like Ha: μ > μ₀ or Ha: μ < μ₀, so the p-value comes from only one tail of the sampling distribution instead of both.
A one-tailed test is a hypothesis test where the alternative hypothesis points in one specific direction. Instead of asking "is the population mean different from μ₀?", you ask "is it bigger?" or "is it smaller?" That choice gets baked in when you write Ha: μ > μ₀ or Ha: μ < μ₀, and it determines exactly how you compute the p-value. With a one-tailed test, the p-value is the probability of getting a test statistic at least as extreme as yours in that one direction, assuming the null hypothesis (H₀: μ = μ₀) is true.
The practical effect is simple. For the same data, a one-tailed p-value is half of a two-tailed p-value, because you're only measuring area in one tail of the t-distribution. That means a one-tailed test has more power to detect an effect in the direction you predicted. The catch is that you have to pick the direction before looking at the data, based on the research question. You don't get to peek at the sample mean and then choose the tail that makes your result look significant.
This term lives in Topic 7.5 (Carrying Out a Test for a Population Mean) in Unit 7, and it sits inside three learning objectives. For AP Stats 7.5.A, the direction of Ha doesn't change the test statistic formula t = (x̄ − μ₀)/(s/√n), but it changes which tail area you shade. For AP Stats 7.5.B, interpreting the p-value correctly means stating it as a probability computed assuming H₀ is true, in the direction your alternative specifies. For AP Stats 7.5.C, your decision (reject or fail to reject H₀ by comparing p to α) depends on a p-value that the tail choice directly controls. The same logic carries through every inference procedure on the exam, including tests for proportions in Unit 6, so getting the one-tailed idea down once pays off everywhere.
Keep studying AP Statistics Unit 7
Alternative Hypothesis (Unit 6 & 7)
The alternative hypothesis IS the switch that makes a test one-tailed or two-tailed. If Ha uses > or <, you have a one-tailed test; if it uses ≠, you have a two-tailed test. Read the research question's verb ("increased," "decreased," "changed") and the tail follows automatically.
P-value (Units 6-7)
The tail choice changes the p-value's size, not its meaning. A one-tailed p-value is the area in one tail beyond your test statistic, which is half the two-tailed value for the same t-score. The interpretation still starts with "assuming the null hypothesis is true."
t-score and Test Statistic (Unit 7)
The t-statistic calculation is identical for one-tailed and two-tailed tests. The standardized distance from x̄ to μ₀ doesn't care about your hypotheses. The directionality only shows up afterward, when you convert that t-score into a tail probability.
Significance Level (Units 6-7)
In a one-tailed test, all of α sits in one tail instead of being split between two. That's why one-tailed tests are more powerful in the predicted direction. A result can clear a one-tailed α = 0.05 cutoff that it would miss two-tailed.
Multiple-choice questions love testing whether you can match a research question to the right alternative hypothesis. If a question says a method "improves scores" or a machine "underfills bottles," that wording demands a one-tailed Ha, and choosing ≠ instead changes the p-value and possibly the decision. Other MCQs hand you a p-value and an α and ask for the decision, like deciding what happens when p = 0.02 and α = 0.01 (fail to reject, since 0.02 > 0.01). On FRQs, the four-step significance test (state, plan, do, conclude) requires you to write Ha with the correct direction in the "state" step. A wrong direction usually costs you the hypotheses point and can wreck your conclusion. Also watch the classic interpretation traps from practice questions, like claiming a small p-value "proves" the alternative hypothesis or saying there's a "0.2% chance the method doesn't work." Both misread what a p-value is.
A one-tailed test asks a directional question (is μ greater than μ₀? is it less?) while a two-tailed test asks whether μ differs from μ₀ in either direction. The test statistic is computed identically, but the two-tailed p-value counts area in both tails, so it's double the one-tailed p-value for the same data. Use the research question's wording to decide. "Increased" or "decreased" means one-tailed; "changed" or "different" means two-tailed. You cannot switch from two-tailed to one-tailed after seeing the data just to shrink your p-value.
A one-tailed test has a directional alternative hypothesis, either Ha: μ > μ₀ or Ha: μ < μ₀, while a two-tailed test uses Ha: μ ≠ μ₀.
The direction comes from the research question's wording, and you choose it before collecting or looking at the data.
The test statistic t = (x̄ − μ₀)/(s/√n) is calculated exactly the same way whether the test is one-tailed or two-tailed.
For the same data, a one-tailed p-value is half the two-tailed p-value because you only measure the area in one tail of the t-distribution.
The decision rule is unchanged. Reject H₀ if the p-value is less than or equal to α, and fail to reject H₀ if the p-value is greater than α.
A small p-value gives evidence for the alternative hypothesis but never proves it, and it is not the probability that H₀ is true.
It's a significance test where the alternative hypothesis specifies a direction, like Ha: μ > μ₀ or Ha: μ < μ₀. The p-value is then the probability, assuming H₀ is true, of getting a result at least that extreme in that single direction.
Look at the alternative hypothesis, which comes from the research question's wording. "Greater," "increased," "more than," or "less than" means one-tailed; "different" or "changed" means two-tailed.
Yes, for the same test statistic. The two-tailed p-value adds up area in both tails of the t-distribution, so it's exactly double the one-tailed value. That's why a result like t = 1.8 might be significant one-tailed at α = 0.05 but not two-tailed.
No. A small p-value means your data would be unlikely if H₀ were true, which is convincing evidence for Ha, but it never proves anything. Saying p = 0.001 "proves" Ha is a classic interpretation error AP graders penalize.
No. The direction of the alternative hypothesis has to be set by the research question before you analyze the data. Choosing the tail after seeing which way your sample mean leans inflates your chance of a Type I error and would lose you credit on an FRQ.