Extra exercise 15
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(same data as in Supplementary Exercise 7.102)

Data: 2 samples of changes (improvements, differences after-before) in
spatial-temporal reading test scores for 34 children attending six
months of piano lessons and 44 children in a control group.

Model: the 2 samples are independent and each a simple random sample 
(i.i.d. sample) from a distribution with unknown mean, median and standard
devation.

When using the Wilcoxon-Mann-Whitney test we may make no further assumptions
about the distributions, and test
  H0: P1=P2, versus Ha: P1<>P2,
where P1 and P2 are the distributions of improvement scores in the two
groups of children (piano lessons and control). The alternative
hypothesis may be expressed that either P1 is systematically larger than
P2, or P2 is systematically larger than P1.

Alternatively, we may make the "delta-assumption" that the two
distributions are of the same shape (only differ by their location), and
test the hypotheses
  H0: median1=median2, versus Ha: median1<>median2

The wording of the problem strongly hints at a one-sided alternative, but
as discussed in the solution for Exercise 7.102, I find little
justification for a one-sided alternative in the description of the data
and problem.

Minitab commands and listing:

---

 WOpen "R:\Chapter 7\ex07_102.mtw".

 Name c4 'change'
 Unstack ('change');
  Subscripts 'group';
  After;
  VarNames.

 Mann-Whitney 95.0 'change_control' 'change_piano';
  Alternative 0.

Mann-Whitney Test and CI: change_control, change_piano 

                 N  Median
change_control  44   0.000
change_piano    34   4.000

Point estimate for ETA1-ETA2 is -3.000
95.1 Percent CI for ETA1-ETA2 is (-5.000,-2.000)
W = 1294.0
Test of ETA1 = ETA2 vs ETA1 not = ETA2 is significant at 0.0000
The test is significant at 0.0000 (adjusted for ties)

Comments:
---------
The Wilcoxon-Mann-Whitney test gives clear evidence that the two distributions
are different, with an approximate P-value below 0.00005. Because the median of
the piano group is higher than the control group, we conclude that the piano
group is generally higher in score than the control group. The precise wording 
hereof is that the distribution for the piano group is systematically larger 
than that of the control group.