Supplementary Exercises 7.58 and 7.59 of IPS7e
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Data: 34 difference scores of preschool children in tests for
spatial-temporal reasoning after and before attending piano lessons.

Model: the 34 observations are a simple random sample (i.i.d. sample)
from a distribution with mean mu and standard devation sigma, both of
which are unknown parameters. The mean mu corresponds to the difference
in mean scores after minus before the piano lessons.

Note that the original scores (before and after the piano lessons)
constitute two paired samples, and the differences therefore become a
single sample


7.58:
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Minitab commands and output:

 Describe 'change';
  Mean;
  SEMean;
  StDeviation;
  QOne;
  Median;
  QThree;
  Minimum;
  Maximum;
  Skewness;
  Kurtosis;
  N;
  NMissing.

Variable   N  N*   Mean  SE Mean  StDev  Minimum     Q1  Median     Q3  Maximum  Skewness
change    34   0  3.618    0.524  3.055   -3.000  2.000   4.000  6.000    9.000     -0.36

Variable  Kurtosis
change       -0.28

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 GSummary 'change'.

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 PPlot 'change';
  Normal;
  Symbol;
  FitD;
  Grid 2;
  Grid 1;
  MGrid 1.
Probability Plot of change 
The P-value of the Anderson-Darling test of normality is 0.227.

---

 OneT 'change';
  Confidence 95.0;
  Alternative 0.

Variable   N   Mean  StDev  SE Mean      95% CI
change    34  3.618  3.055    0.524  (2.552, 4.684)

Comments:
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The distribution looks quite symmetric, but also quite discrete (takes only
few distinct values). The normal plot looks excellent, so a normal
distribution would seem ok; also, the P-value of the normality test is clearly 
above 0.05. Therefore, the 95% confidence interval for the mean can be assumed 
to be accurate. The mean difference is positive, and the 95% CI does not include 0.
Therefore, there seems to be some improvement in the scores from before to
after the piano lessons.


7.59:
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The null hypothesis is  H0: mu=0
We may take the alternative hypothesis one-sided, Ha: mu>0, 
because "neurobiological arguments" suggest that there should be an 
improvement. (It would also be completely ok to take a two-sided
alternative.)

 OneT 'change';
  Test 0;
  Confidence 95.0;
  Alternative 1.

Test of mu = 0 vs > 0

Variable   N   Mean  StDev  SE Mean  95% Lower Bound     T      P
change    34  3.618  3.055    0.524            2.731  6.90  0.000

Comments:
--------
The test statistic and P-value are displayed above. DF = n-1 = 33.
By the very low P-value we conclude that there is very strong evidence
of an improvement in the children's reasoning scores, from before to after
the piano lessons. The P-value may be considered exact, because the
assumption of a normal distribution seems reasonable for these data. The
confidence interval from 7.58 gave the most likely range of improvement
as roughly 2.5-4.5 units (of reasoning scores). Whether this is a
magnitude of practical interest/importance we cannot say from these
data.