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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."

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  • $\begingroup$ The article has 100(1 - 2 alpha) as Antoni writes in his answer. But you confusingly have (incorrectly) alpha/2 in your question. $\endgroup$ – PatrickT Apr 1 '18 at 21:13
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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.

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  • $\begingroup$ 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? $\endgroup$ – rnorouzian Jan 24 '17 at 18:56
  • $\begingroup$ 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. $\endgroup$ – Antoni Parellada Jan 24 '17 at 18:58
  • $\begingroup$ If you want to get that same area (with its cutoff point on the $x$-axis by "cheating" and actually getting two red areas... $\endgroup$ – Antoni Parellada Jan 24 '17 at 18:59
  • $\begingroup$ one at either end of the curve (a $\text{mean}\pm \text{1/2 CI}$)... $\endgroup$ – Antoni Parellada Jan 24 '17 at 19:00
  • $\begingroup$ ... still preserving the original $5\%$ red area that you had at the beginning, you will have to double $\alpha,$ $\endgroup$ – Antoni Parellada Jan 24 '17 at 19:01

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