# Confidence Interval vs. One sided Hypothesis Test

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## Summary:

Must a one sided hypothesis test agree with the confidence interval of the same level of significance?
I learned that the confidence interval and a two sided hypothesis test (test for difference) agree if both have the same significance level.
Is that also true for one sided hypothesis tests? For instance, must a right-tailed Hypothesis test with alpha = 0.05 agree with the 95% confidence level centered at the estimator? I'm thinking if alpha is 0.05 then the tail areas in a confidence interval would be 0.025. But in the right sided hypothesis test the right tail area of alpha is 0.05 and not 0.025.

## Answers and Replies

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andrewkirk
Homework Helper
Gold Member
In the two sided case, with two-sided confidence interval level ##\alpha## they agree in the sense that a two-sided test at level ##1-\alpha## of the null hypothesis that the two means are the same will reject the null hypothesis if and only if the two-sided ##\alpha##-level confidence interval for the difference of means, constructed by the same methodology, does not contain zero.

In the one sided case, with one-sided confidence interval level ##\alpha## they agree in the sense that a one-sided test at level ##1-\alpha## of the null hypothesis that the first mean is no greater than the second will reject the null hypothesis if and only if the right-tailed, level-##\alpha## confidence interval for first mean minus second mean, constructed by the same methodology, does not contain zero.

Note that in the one-tailed case, the confidence interval must also be one-sided, which means it is of type ##(a,\infty)## or ##(-\infty, b)##. For a two-sided confidence interval, there will not be equivalence in any useful sense.

Homework Helper
Remember why (traditional) two-sided CIs are equivalent to traditional hypothesis tests (for the same $\alpha$).
In a two sided test (I'm using a t-test outline here, but work is identical in organization for many of the basic tests), if the null hypothesis is not rejected, all of the following statements are equivalent.

\begin{align*} & -z \le \dfrac{\bar{x} - \mu_0}{\frac s{\sqrt n}} \le z \\ & \Leftrightarrow \\ & -z \dfrac s{\sqrt{n}} \le \bar{x} - \mu_0 \le z \dfrac s{\sqrt n} \\ & \Leftrightarrow \\ & -\bar{x} - z \dfrac{s}{\sqrt n} \le -\mu_0 \le -\bar{x} + z \dfrac{s} {\sqrt{n}} \\ & \Leftrightarrow \\ & \bar{x} - z \dfrac s {\sqrt{n}} \le \mu_0 \le \bar{x} + z \dfrac s {\sqrt n} \end{align*}

so that the null hypothesis isn't rejected if and only if the true parameter value is in the two-sided confidence interval.

If you try the same organization of work to go from not rejecting the null hypothesis when the
alternative is ## H_0 \colon \mu > \mu_0## you'll find that there is no introduction of the two-sided interval.