Testing Hypotheses
About a
Difference of Two Means

where

Notes:

• As usual, you may approximate the t percentiles (if necessary) by the corresponding standard normal percentiles when the degrees of freedom is sufficiently large.
• When the variances are known (not necessarily equal), use the following test statistic
  
  
  in the various tests given above and replace the t percentiles by the corresponding standard normal percentiles. The resulting tests are still applicable when the normality assumption(s) do not hold provided sample sizes are large enough.
• When data comes from a paired comparsion study, we can use the tests for a single mean given earlier to obtain tests for the difference of two population means. To do so, we compute the differences
  
  D_i = X_i - Y_i,    i = 1,...,n,
  and convert our problem to one of testing hypotheses about the common mean of the D_i's. This common mean is equal to the difference of two means of our original problem.

Examples:

• See Examples 5, 6, 9 and 10 in Johnson and Bhattacharyya (1996), pp. 406-416 tests based on independent random samples.
• See Example 12 in Johnson and Bhattacharyya (1996), pp. 431-433 for a test based on paired samples.