One-Tailed or Two-Tailed Hypothesis Test?

Pick the Right Hypotheses

With a one-tailed test, say for μ<4.5, you’re saying that you consider “equal to 4.5” and “greater than 4.5” the same thing, that if μ isn’t less than 4.5 then you don’t care whether it’s equal or greater. Sometimes you really don’t care, but very often you really do. If the problem statement is ambiguous, or if this is real life and you have to do a hypothesis test, how do you decide whether to do a one-tailed or two-tailed test?

Testing two-tailed keeps you honest. Do a two-tailed test unless you can honestly say, without looking at the data, that only one direction of difference matters.

For example, suppose the existing drug cures people in an average of 4.5 days, and you’re testing a new drug. If you test for μ<4.5, you’re saying that it doesn’t matter whether the new drug takes the same time or more time, but of course it matters very much. What you want to test is whether the new drug is different (μ≠4.5). Then if it’s different, you can conclude whether it’s faster or slower.

Another way to look at this whole business: a one-tailed test essentially doubles your α — you’re much more likely to reach a conclusion with dicey data. But that means double the risk of being wrong with a Type I error — not a good thing!

There are two main contexts in which a one-tailed test is appropriate: “(a) where there is truly concern for the outcomes in one [direction] only and (b) where it is completely inconceivable that the results could go in the opposite direction.” (Dubey, cited on page 132 of Kuzma and Bohnenblust, Basic Statistics for the Health Sciences, fifth edition)

Sometimes the same situation can call for a different test, depending on your viewpoint. Here’s an example.

Suppose you’re the county inspector of weights and measures, checking up on a dairy and its half gallons of milk. Legally, half a gallon is 64 fluid ounces. To an inspector, “Dairylea gives 64.0 ounces in the average half gallon” and “Dairylea gives more than 64.0 ounces in the average half gallon” are the same (legal), and you care only about whether Dairylea gives less (illegal). A one-tailed test (<) is correct.

But now shift your perspective. You’re Dairylea management. You don’t want to short the customers because that’s illegal, but you don’t want to give too much because that’s giving away money. You make a two-tailed test (≠).

p < α in Two-Tailed Test: What Does it Tell You?

After a two-tailed test, if p<α then you can interpret the result as one-tailed. Suppose you did test a new drug against H₁ of μ≠4.5, and 100 people had an average recovery of 4.2 days with s = 1.0 day. Your p-value is 0.0034, and you reject H₀ and accept H₁. You conclude that the recovery time is different for the new drug.

But you can go further and say that the recovery time is less with the new drug. You do this by combining the facts that (a) you’ve proved recovery time is different, which means it must be either less or more, and (b) the sample x̅ was less than μ_o. This is legitimate because if you went back and tested against μ<4.5 your p-value would be even smaller than 0.0034.

You can phrase your conclusion something like this, first answering the original question then going beyond it: “At the 0.01 level of significance, mean recovery time with the new drug will be different from 4.5 days — in fact, it will be less than 4.5 days.”