Confidence Interval
for a
Difference of Two Means

where

and
with n₁+n₂-2 degrees of freedom.

Notes:

• With the exception of the pooled sample standard deviation S_p, subscripts for sample statistics in the above formulae match those for the corresponding normal distribution.
• Limits of the above confidence interval may be expressed as
  
  
  As before and if necessary, we can approximate the t percentile in the above confidence limits by the corresponding standard normal percentile when the degrees of freedom is sufficiently large.
• When the variances are known (whether equal or not), use the confidence limits
  
  where
  
  
  We can relax the assumption of normality when sample sizes are sufficiently large and use the above confidence limits to obtain approximate confidence intervals.
• When data comes from a paired comparsion study, we can use the confidence interval formula for a single mean given earlier to obtain a confidence interval for the difference of two population means. To do so, we compute the mean and standard deviation of the differences
  
  D_i = X_i - Y_i,    i = 1,...,n,
  and then use these summary statistics in place of the corresponding ones in the abovementioned formula.

Examples:

• See Examples 4, 8 and 11 in Johnson and Bhattacharyya (1996), pp. 405-417 for confidence intervals based on independent random samples.
• See Example 12 in Johnson and Bhattacharyya (1996), pp. 431-433 for a confidence interval based on paired samples.