e-Statistics

One-sided CI

If you are interested in the confidence interval bounded only from the above or the below, you need one-sided confidence intervals for the population mean $\mu$ . Suppose the sample mean $\bar{X}$ = and the sample standard deviation S = from data of sample size n = . With the significance level $\alpha$ = , we can obtain the $(1-\alpha)$ % one-sided confidence interval only with upper or lower bound.

One-sided with upper bound	$\displaystyle\bigg(-\infty,\: \bar{X} + t_{\alpha,n-1}\frac{S}{\sqrt{n}}\bigg) = (-\infty,$ )
One-sided with lower bound	$\displaystyle \bigg(\bar{X} - t_{\alpha,n-1}\frac{S}{\sqrt{n}},\:+\infty \displaystyle\bigg)$ = ( $,\:+\infty)$

If the standard deviation $\sigma$ is known, we set $S = \sigma$ and replace the critical point $t_{\alpha,n-1}$ with $z_{\alpha}$ = of the standard normal distribution.

There is an interesting relationship between confidence intervals (CI's) and hypothesis tests: If the null hypothesis is rejected with significance level $\alpha$ then the corresponding CI (see the table below) with confidence level $(1-\alpha)$ does not contain the value $\mu_0$ targeted in the hypotheses, and vice versa. Therefore, it is often reasonable to present the CI suggested in the following table when the null hypothesis is rejected.

Hypothesis test	$(1-\alpha)$ -level confidence interval
$H_A: \mu \neq \mu_0$	Two-sided
$H_A: \mu < \mu_0$	One-sided with upper bound
$H_A: \mu > \mu_0$	One-sided with lower bound