# Alternative hypothesis when null hypothesis is ambiguous?

This is more of a textbook semantics issue. Is the alternative hypothesis always two-tailed when all that is known is a null hypothesis $H_0$ where $p$ equals some arbitrary figure, where an arbitrary sample proportion and significance level are also given (assuming all requirements are met)?

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When your H0 is "p=something" and your Ha is "p≠something", then yes, a two-tailed test is what's needed. "One tailed" replaces the "≠" with ">" or "<", whichever is appropriate. –  non-expert Jan 4 '11 at 5:16

## 2 Answers

Yes, although it's more than just a textbook semantics issue. In the absence of information about the form the alternate hypothesis should take (e.g., in the absence of a research question like "Does the new drug work better than the best-known drug on the market?") then the formulation of the alternate should just be the negation of the null. Thus the form of $H_0$ and $H_a$ should be $p = p_0$ and $p \neq p_0$, respectively. As non-expert points out in the comments, this then means that you need to perform a two-tailed test.

Two comments:

1. Remember that the statements of the null and alternate hypotheses don't have anything to do with the sample proportion or with the significance level. In particular, it's good practice to formulate the hypotheses before the results from the sample are known.

2. It is interesting that Ronald Fisher, one of the developers of tests of significance, effectively took the position that one should always use the two-sided alternate hypothesis. To quote from the Wikipedia article on the alternative hypothesis, "[The alternate hypothesis] was not part of Ronald Fisher's formulation of statistical hypothesis testing, and he violently opposed its use. In Fisher's approach to testing, the central idea is to assess whether the observed dataset could have resulted from chance if the null hypothesis were assumed to hold, notionally without preconceptions about what other model might hold. Modern statistical hypothesis testing accommodates this type of test since the alternative hypothesis can be just the negation of the null hypothesis."

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This was very informative, thank you. Negation is definitely the logical and less presumptuous decision. Your first comment did it for me since I was wondering if the test statistic could tell me anything about what type of test should be used--which it can't. –  user2054 Jan 5 '11 at 1:09

The decision as to how many tails to include depends upon the relative losses associated with the potential decisions. If testing a new process (or drug) against an old method or formulation, the test should always be one-sided.

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