The Professor's Response
Is it safe to assume that in HW: 9.5, where The Goverment's mean is 35.50 and the Private is 54.6, that the final interval is going to be negative. (This could be in any situation) Per say if Goverment mean is 54.6 and 35.50 that the final interval is positive ?
See the professor's answer below.
No, you can't assume the interval will be all positive or all negative just by looking at the sample means. The first reason is that it depends on how you set up the subtraction in the interval. The second reason is that there may not be a significant difference between the two means in question. In that case, you will get an interval that is both negative and positive.
There is no way to avoid actually calculating the confidence interval to determine if there is a significant difference between the means.
Hope that answers your question,