e-Statistics > 4480-5480 Probability and Statistics II

Confidence Intervals

A confidence interval (CI) provides a range of plausible values for the unknown population mean $ \mu$. The choice of the confidence level $ (1-\alpha) =$ is typically 90%, 95% or 99%, and represents the chance that the CI does indeed contain the true population mean $ \mu$. The construction of CI is based upon the sample mean $ \bar{X} =$ and the sample standard deviation $ S =$ from data of sample size $ n =$ . The most commonly used confidence interval is a two-sided CI which is centered at the mean $ \bar{X}$ and extends either side an equal amount.

$ \displaystyle\bigg(\bar{X} - t_{\alpha/2,n-1}\frac{S}{\sqrt{n}} =$ , $ \displaystyle\bar{X} + t_{\alpha/2,n-1}\frac{S}{\sqrt{n}} =$ $ \displaystyle\bigg)$

If the variance $ \sigma^2$ is known and equal to $ S^2$, the critical point $ t_{\alpha/2,n-1}$ can be replaced by $ z_{\alpha/2}$ of the standard normal distribution.

$ \displaystyle\bigg(\bar{X} - z_{\alpha/2}\frac{\sigma}{\sqrt{n}} =$ , $ \displaystyle\bar{X} + z_{\alpha/2}\frac{\sigma}{\sqrt{n}} =$ $ \displaystyle\bigg)$

For example, in order for vaccine to be approved for widespread use, it must be established that the probability $ p$ of serious adverse reaction must be less than $ p_0$. In this case we should set `` $ H_0:\: p \ge p_0$ versus $ H_A:\: p < p_0$'' to see whether we can reject $ H_0$ in favor of $ H_A$.

Let $ X =$ be the frequency of the specified type in categorical data of size $ n =$ . For example, in order for vaccine to be approved for widespread use, we must estimate the probability $ p$ of serious adverse reaction. In this example $ X$ will be the number of participants who suffered adverse reaction among $ n$ participants. Then the two different formulas

$ \displaystyle
\hat{p} \pm z_{\alpha/2}\sqrt{\frac{\hat{p}(1-\hat{p})}{n}} =$ ( , )

$ \displaystyle
\frac{n\hat{p} + z_{\alpha/2}^2 / 2
\pm z_{\alpha/2} \sqrt{n\hat{p}(1-\hat{p}) + z_{\alpha/2}^2/4}}{n+z_{\alpha/2}^2}
=$ ( , )

are available for the confidence interval with level $ (1-\alpha) =$ . Although the first formula is easier to calculate, the second is known to be more accurate and widely used.