Assignment 15b
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Spearman rank correlation coefficient
   (non-parametric counterpart to the Pearson correlation coefficient)

Wilcoxon Symmetry Test (aka Wilcoxon Signed Rank Test)
   (non-parametric counterpart to the t-procedure for paired data)

Mann-Whitney U-Test (aka Wilcoxon Rank-Sum Test)
   (non-parametric counterpart to the t-procedure for independent samples)

Subjects are asked to pronounce a large number of compound words 
which are later analysed.  The compounds can take one of two forms,
the first of which is 

    ... + vowel + t + s + word-boundary + p + vowel + ..., e.g., fietspad

The second form is:

    ... + vowel + t + s + word-boundary + b + vowel + ..., e.g., fietsband
 
In total 112 compounds are pronounced, 55 of the first sort and 57 of
the second.  The length (in ms) of [t] and [s] is measured and recorded.

We will compare the respective lengths of [t] and [s], asking whether
the lengths are systematically related, and we also compare the
lengths of [s] in the first sort of compund with the lengths in the
second sort of compound, asking whether the lengths are different. .
(Source: research by Wouter Jansen, Linguistics, Rijksuniversiteit
Groningen.)

The data may be found in z:\public\share\heeringa\samen.txt.  Read
this ASCII file into S+. Define the three columns of the table and
name the columns 'form', 't_len' en 's_len'.

a. When is it preferable to use the Spearman rank correlation
   coefficient instead of the Pearson correlation coefficient?

b. Formulate H_0 en H_a for the question as to whether the lengths of
   [t] and [s] are related.  Computer the correlation coefficient rho
   between the length of the [t] and the length of the [s], the
   t-value and the p-value.  Is there evidence for correlation?

c. Formulate H_0 en H_a for the question about the average lengths of
   [t] as opposed to [s].  Do not distinguish the two types of
   compounds in asking this question.  The Wilcoxon symmetry test may
   be used as long as the distribution of the two groups is
   symmetrical.  Draw a histogram showing the lengths of [t] and a
   second showing the lengths of [s].  May we apply the Wilcoxon
   symmetry test?

d. Use the Wilcoxon symmetry test to test the hypotheses in (c).  Are
   the lengths significantly different?

e. We turn to the question of whether [s] is the same length in the
   two sorts of compounding.  Formulate the appropriate H_0 and H_a.
   The Mann-Whitney U-test may be used if the distributions of the two
   groups have roughly the same form.  Draw histograms for the lengths
   of [s] in the two different sorts of compounds.  May we apply the
   Mann-Whitney U-test here?

f. Apply the Mann-Whitney U test.  Is there a significant differnce
   between the length of [s] in the two sorts of compound?