## Confidence Intervals

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

If the variance is known and equal to , the critical point can be replaced by of the standard normal distribution.

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

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

are available for the confidence interval with level . Although the first formula is easier to calculate, the second is known to be more accurate and widely used.