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 from data of sample size . With the confidence level $(1-\alpha) =$ , we can obtain the one-sided CI only with upper or lower bound.

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

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
$H_0: \mu = \mu_0$ vs. $H_A: \mu \neq \mu_0$	Two-sided
$H_0: \mu \ge \mu_0$ vs. $H_A: \mu < \mu_0$	One-sided with upper bound
$H_0: \mu \le \mu_0$ vs. $H_A: \mu > \mu_0$	One-sided with lower bound