Jain: Confidence Intervals

Parameter vs. statistic
Central limit theorem. If are independent samples from a distribution with mean and standard deviation , then

Hence, a convidence interval for is given by

where is the -quantile of N(0,1). See Tables A.[23].
For small sample size (e.g., n less than 30), the central limit theorem does not hold. If the population is normally distributed, one can still use the t-distribution to construct a confidence interval for the mean, given the sample mean and sample standard deviation.
How to compare two alternatives:
- If the samples are paired, just compute the pairwise differences and test to see if the mean difference is significantly different from zero.
- If the samples are unpaired, we can
  - Compute confidence intervals for each sample and see if they overlap; or
  - Use a t-test as described in Section 13.4.2.