10. Comparing Two Population Means
The form of the confidence interval is
but, as with hypothesis testing, we have two cases to choose from to get the formula for :
Case 1 : Variances of the 2 populations unequal}
where the degrees of freedom to use when looking up in the t Distribution Table is
Case 2 : Variances of the 2 populations equal
where we use
when looking up .
To select the appropriate formula for we need to do a preliminary hypothesis test on . An odd combination of hypothesis test followed by confidence interval calculation.
Insight! By now you should have noticed that the formulae for are just times standard error of the mean. This whole -transformation thing should be becoming somewhat transparent.
Example 10.6 : Find the 95 confidence interval for for the data of Example 10.4 :
First use -test to see which formula to use. We did this already in Example 10.4 (the data come from that question) and found that we believed with .
Next, look up in the t Distribution Table for 95 confidence interval for :
be careful of the order!