<p>It seems to me reading the comments to the original post that people got distracted by issues which were possibly not core to your question. Looking at the summary you give in your post here it seems to me that if I take the null and alternative hypotheses and the paragraph which immediately follows them it might be summarised as:</p>
<blockquote>
<p>Is it meaningful to compare the overall mean with the mean of one of the sub-groups which comprise it and if it is, is that the same as testing for equality of the sub-group means?</p>
</blockquote>
<p>If that is what you ask then perhaps it might be better to ask that as a new question.</p>
<p>There is also the issue of, if it is possible, how to program a computer to do it but as I am sure you know including that is liable to get people to vote to close it as off-topic so I would strongly advise resisting any temptation to include that.</p>
<p>Of course if I have mis-interpreted your question then that may help you to see how to re-word it to express what you do what to ask. Just for the record I had not seen the question before I saw this post on meta.</p>
