Supplementary Exercise 10.40 of IPS7e
-------------------------------------

Association between personality traits and GRE (Graduate Record 
Examination) scores for 342 persons. It is not stated explicitly what
what the shown correlations are for, i.e. what the actual variables
were. Presumably the respondents answered lots of questions that
allowed the researchers to compute a score for both each respondent's
strength in analytical, quantitative and verbal domains, and for the
listed personality traits. That is, each respondent would have a score
for each of the 9 variables (6 traits, 3 domains), and the tables gives
the correlations between some of these variables across the 342 persons.

The statistical model assumed for each pair of variables is either a sample 
from a joint normal distribution or a regression of GRE scores on personality 
traits, with normally distributed errors about the regression line. The first 
model may seem most natural from the information we have about the data. There
is nothing to indicate that some of the variables were (partly)
controlled, so computing correlations between them should be fine.

The significance test for H0: rho=0, or equivalently for H0: beta1=0 in
the linear regression, is computed by the formula
  t = r*sqrt((n-2)/(1-r*r))
and the t-statistic is assessed in a t-distribution (n-2). Due to the
very large n, we could use instead a standard normal distribution. For
both tests, a two-sided alternative hypothesis should be used.

Although both the PSLS and S textbooks have statistical tables for the
correlation coefficient, these are of limited use here because our
degrees of freedom (342-2=340) are not included in the table. We will 
instead compute the t-statistic and its P-value manually, using Minitab.
Although there are many values, it turns out that we only need to
compute the P-value for a few of them to get the desired information 
about significance.

Minitab commands for (manual) computation, after the 18 correlations
have been typed into column c3:

---

 Name c1 "personality"
 TSet 'personality'
  3(  "conscientiousness" "rationality" "ingenuity" "quickness" "creativity" "depth" )1
  End.

 Name c2 "GREdomain"
 TSet 'GREdomain'
  1(  "analytical" "quantitative" "verbal" )6
  End.

 Name c3 "rho"
 Set 'rho'
  1( -0.17 -0.06 -0.06 0.21 0.24 0.06 -0.14 -0.03 -0.08 0.15 0.26 0.08 -0.12 -0.08 -0.02 0.26 0.29 0.15 )1
  End.

 Name C4 't'
 Let 't' = 'rho'*sqrt(340/(1-'rho'^2))
 Let c5 = -abs('t')

 CDF C5 c6;
  T 340.

 Name C7 'P'
 Let 'P' = 2*c6
 Print 'personality' 'GREdomain' 'rho' 't' 'P'.

Data Display 

Row  personality        GREdomain       rho         t         P
  1  conscientiousness  analytical    -0.17  -3.18095  0.001603
  2  rationality        analytical    -0.06  -1.10834  0.268497
  3  ingenuity          analytical    -0.06  -1.10834  0.268497
  4  quickness          analytical     0.21   3.96052  0.000091
  5  creativity         analytical     0.24   4.55862  0.000007
  6  depth              analytical     0.06   1.10834  0.268497
  7  conscientiousness  quantitative  -0.14  -2.60715  0.009532
  8  rationality        quantitative  -0.03  -0.55342  0.580338
  9  ingenuity          quantitative  -0.08  -1.47987  0.139834
 10  quickness          quantitative   0.15   2.79751  0.005443
 11  creativity         quantitative   0.26   4.96491  0.000001
 12  depth              quantitative   0.08   1.47987  0.139834
 13  conscientiousness  verbal        -0.12  -2.22880  0.026481
 14  rationality        verbal        -0.08  -1.47987  0.139834
 15  ingenuity          verbal        -0.02  -0.36886  0.712465
 16  quickness          verbal         0.26   4.96491  0.000001
 17  creativity         verbal         0.29   5.58745  0.000000
 18  depth              verbal         0.15   2.79751  0.005443

Comments:
---------
It is seen that for the correlation to be significant at P<0.05, its
value needs to be numerically larger than 0.08. If we want to adjust
for carrying out 18 tests, we can use a Bonferroni procedure, by which
we would require P-values less than 0.05/18=0.0028 for significance. In
the table above, all correlations numerically between 0.12 and 0.15 
would become non-significant after such an adjustment, and P-values
numerically above 0.15 would remain significant.

The main conclusions based on significant correlations would be that
only conscientiousness is negatively associated with GRE scores.
Positive associations with test scores have quickness, creativity and
(for verbal scores only) depth personality traits.

These findings do not agree with statements that the tests penalize deep
thinkers and creative persons, because both of these are positively
associated with scores. The findings do on the other hand agree with
statements that working quickly is beneficial with these tests, because
quickness is positively associated with test scores, and because
conscientiousness (which presumably is a slower process) is negatively
associated with test scores.