What is the most Most relevant statistic method to compare those samples? One person is answering a survey. There are two type of questions :


*

*4 questions about intellectual work affinity.

*4 questions about manual work affinity.


For each of these question, the person gives an aswer from 1 (do not agree) to 5 (totally agree).
I want to know if there is a significant difference between answers about intellectual work and manual work, and eventually be able to say "this individual is more an intellectual / a manual" or "the difference is not significant.
I am thinking of Wilcoxon method, but I would like to have a second opinion as I'm really just learning statistics by myself.
 A: I saw your question just after it was posted. My first thoughts were
that a satisfactory test is not going to be easy to find. I have
thought about it during the day, and I still see serious difficulties,
but by not maybe I can explain my reservations more clearly. And perhaps
suggest one workable method for you to try.
1) I think it is going to be difficult to frame eight Likert-scale
questions that allow subjects to show their views on intellectual
and manual work activities. But let's suppose that is possible.
2) To evaluate one person's view, it would be difficult to make good
judgments based on a rank-based test such as a Wilcoxon test. While
rank-based tests do not assume normal data, they do assume that the
observations are on a continuous scale. For discrete data such as
yours there are likely to be ties, which make it problematic to
compute a P-value. 
For large samples, the Wilcoxon test statistics are nearly normal.
For eight observations nomality would not apply, and some sort of
adjustment (different depending on text or software package) would be
used. Even so, if someone marked four 5's for the intellectual questions and
four 1's for manual work questions, results from the software packages
I tried show a small P-value, suggesting a rejection of the null
hypothesis that the two types of activity are equally preferred, and
giving a warning that the P-value is not 'exact' because of ties.
3) Suppose you assign scores 1 through 5 for answers 1 through 5 on the
intellectual questions and scores -1 through -5 for answers 1 through 5 on
the manual work questions. And then take the sum $S$ of the eight scores as
a test statistic. (You would have to ponder whether this kind
of scoring matches your intent in asking the questions.)  Then scores
could be as low as $-15$ and as high as $15$. 

In a simulation of 100,000 subjects choosing eight possible scores totally
at random (one model for people with no preference for either type of
activity or, perhaps, just not taking your questions seriously), the
distribution of $S$ turned out to be very nearly normal with mean $0$
and SD about $4.$ So you might judge those with $S \ge 8$ as
intellectually inclined, those with $S \le -8$ as inclined toward
manual labor and those in the middle as not showing a significant result.
My simulation involved only 'subjects' with no particular views or
ones not paying much attention to the questions. Among such subjects
about 3% are (perhaps wrongly) classified as preferring manual work
and another 3% classified as preferring intellectual pursuits. 
Among subjects with strong views one way or the other, you might find
much higher percentages with significant results. However, you have
to realize that you may not get very good information from only eight
questions $-$ no matter how carefully crafted.
