## Confidence Intervals

#### ^Basic Principles

The sample mean is a point estimate of the population mean, i.e, it is a single value which we use to represent the population mean. However, the sample mean varies in repeated samples from the population and thus we need to assess (probabilistically) how close the sample mean is to the population mean.

The Confidence Interval applet below illustrates how the CI width depends on various factors. The user has animation controls to change the values of the population mean, population standard deviation, the sample size, and the level of confidence.

The display has 100 horizontal lines representing the computed confidence intervals from m = 100 samples, each of size n (m is used to represent the simulation size, whereas n represents the number of observations in each trial). Each interval is computed as: sample mean ± t x se, where t is the 1 - (alpha/2) quantile of a t distribution with n-1 degrees of freedom and se is the standard error (i.e, s/sqrt(n)). The lengths of the confidence intervals vary since they are based on randomly selected samples (from the same normal parent distribution). Thus the sample mean and standard deviation vary (but not the t value). The confidence intervals enclosing the true population mean (denoted as a red vertical line) are drawn in blue, whereas the intervals not containing the population mean are drawn in yellow.

Example #1

#### ^Exercises

Exercise #1 examines how various factors affect the confidence interval width for mean diameter thickness.

Exercise #2 examines how the sample size, the variability, and the level of confidence affect the confidence interval width for mean diameter thickness.