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 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 $ H_0$ 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