10. Comparing Two Population Means

# 10.5 Confidence Intervals for the Difference of Two 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 :

Solution :

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 :

Compute

So

be careful of the order!