# Matching Confidence limits with One-Sided Hypothesis tests

I'm trying to understand (esp. the boldfaced part) how THIS ARTICLE (see p. 174, top-right corner) is suggesting that:

"to use the confidence intervals to test a statistical hypothesis and to maintain a Type I error rate at alpha:

1. When testing a two-sided hypothesis at the alpha level, use a 100(1 - alpha)% confidence interval.

2. When testing a one-sided hypothesis at the alpha level, USE a 100(1 - alpha / 2)% confidence interval."

• The article has 100(1 - 2 alpha) as Antoni writes in his answer. But you confusingly have (incorrectly) alpha/2 in your question. Apr 1, 2018 at 21:13
• A citation for the article which the OP mentions: Steiger JH. Beyond the F test: Effect size confidence intervals and tests of close fit in the analysis of variance and contrast analysis. Psychol Methods. 2004 Jun;9(2):164-82. doi: 10.1037/1082-989X.9.2.164. PMID: 15137887. Oct 6, 2022 at 9:35

If you generate the two-sided confidence interval with a confidence level of $95\%$ (or $\alpha_1 = 5\%$), the cut-off points (or endpoints) of the interval will leave out a probability of a type I error of $\frac{1}{2} \alpha=2.5\%$ on either end.

If you are performing a one-sided test, and want to preserve a risk $\alpha = 5\%$ of rejecting the null when it is in fact true, you will want to generate a two-sided CI with and confidence level of $90\%$ to leave $5\%$ probability at each end.

So you double the initial $\alpha_1 =5\%$ to $\alpha_2=2\alpha_1=10\%.$

Hence the quote:

1. When testing a two-sided hypothesis at the alpha level, use a $100(1 - \alpha )\%$ confidence interval.
2. When testing a one-sided hypothesis at the alpha level, use a $100(1 - 2 \alpha)\%$ confidence interval. • Antoni, I'm a bit confused. Isn't it true that when we do a one-sided test, we have one-rejection region? If yes, where do we place our rejection region under the sampling distribution? Jan 24, 2017 at 18:56
• Yes. It is true. Just imagine the area under that rejection level in red. It contains, say $5\%$, which is the probability of rejecting the null when it is true. Now picture the other end of the "bell curve" without any areas in red. Jan 24, 2017 at 18:58
• If you want to get that same area (with its cutoff point on the $x$-axis by "cheating" and actually getting two red areas... Jan 24, 2017 at 18:59
• one at either end of the curve (a $\text{mean}\pm \text{1/2 CI}$)... Jan 24, 2017 at 19:00
• ... still preserving the original $5\%$ red area that you had at the beginning, you will have to double $\alpha,$ Jan 24, 2017 at 19:01