Why do we use a one-tailed test [F-test] in analysis of variance (ANOVA)?
F tests are most commonly used for two purposes:
in ANOVA, for testing equality of means (and various similar analyses); and
in testing equality of variances
Let's consider each in turn:
1) F tests in ANOVA (and similarly, the usual kinds of chi-square tests for count data) are constructed so that the more the data are consistent with the alternative hypothesis, the larger the test statistic tends to be, while arrangements of sample data that looks most consistent with the null corresponds to the smallest values of the test statistic.
Consider three samples (of size 10, with equal sample variance), and arrange them to have equal sample means, and then move their means around in different patterns. As the variation in the sample means increases from zero, the F statistic becomes larger:
The black lines ($|$) are the data values. The heavy red lines are the group means.
If the null hypothesis (equality of population means) were true, you'd expect some variation in sample means, and would typically expect to see F ratios roughly around 1. Smaller F statistics result from samples that are closer together
than you'd typically expect. so you aren't going to conclude the population means differ.
That is, for ANOVA, you'll reject the hypothesis of equality of means when you get unusually large F-values and you won't reject the hypothesis of equality of means when you get unusually small values (it may indicate something. but not that the population means differ).
Here's an illustration that might help you see that we only want to reject when F is in its upper tail:
2) F tests for equality of variance*. Here, the ratio of two sample variance estimates will be large if the numerator sample variance is much larger than the variance in the denominator, and the ratio will be small if the denominator sample variance is much larger than variance in the numerator.
That is, for testing whether the ratio of population variances differs from 1, you'll want to reject the null for both large and small values of F.
* (Leaving aside the issue of the high sensitivity to the distributional assumption of this test (there are better alternatives) and also the issue that if you're interested in suitability of ANOVA equal-variance assumptions, your best strategy probably isn't a formal test.)