Extra exercise 8
----------------

(a)
In the Minitab code below we generate 3 columns of length 100, so that
we can compare the findings from 3 replications of the procedure. Before
doing that we set the seed (or base, in Minitab terminology). You need
to use the same seed if you want to get exactly the same results.

Minitab commands and output:

MTB > Base 140919.
MTB > Random 100 c1 c2 c3;
SUBC>   Normal 0.0 1.0.

MTB > Describe C1 C2 C3;
SUBC>   Mean;
SUBC>   SEMean;
SUBC>   StDeviation;
SUBC>   QOne;
SUBC>   Median;
SUBC>   QThree;
SUBC>   Minimum;
SUBC>   Maximum;
SUBC>   Skewness;
SUBC>   Kurtosis;
SUBC>   N.
Descriptive Statistics: C1, C2, C3 

Variable    N     Mean  SE Mean   StDev  Minimum       Q1   Median      Q3  Maximum
C1        100   0.0252   0.0925  0.9247  -2.2454  -0.6019   0.0748  0.6113   2.2247
C2        100  -0.0153   0.0990  0.9900  -2.2387  -0.6915  -0.1234  0.7042   1.9063
C3        100    0.019    0.106   1.055   -2.068   -0.761    0.112   0.787    2.748

Variable  Skewness  Kurtosis
C1            0.00     -0.06
C2           -0.09     -0.55
C3            0.08     -0.16

MTB > GSummary C1 C2 C3.
Summary Report for C1 
Summary Report for C2 
Summary Report for C3 

MTB > PPlot C1 C2 C3;
SUBC>   Normal;
SUBC>   Symbol;
SUBC>   FitD;
SUBC>   Grid 2;
SUBC>   Grid 1;
SUBC>   MGrid 1;
SUBC>   Panel.
Probability Plot of C1, C2, C3 


Comments:
---------
All 3 columns generated seem to conform well with the standard normal
distribution; note that the skewness and kurtosis are both close to zero. 
The histograms are not perfect but deviations from the
normal curve are only minor. All normal probability plots are reasonably
straight lines, with only a few points off. The normality tests are all far
from significant. Try instead with the seed (base) set at 140920 to see
some less nice results. Both sets of results are evidently valid
simulations from a standard normal distribution.

The reason that the distributions look so good is that with 100 points
the approximation of the data to the normal distribution is quite good.
Try repeating with n=30 observations...


(b) 
We use the same approach in Minitab as described above, with simulated
data from a uniform distribution (0,1).

Minitab commands and output:

MTB > Base 140919.
MTB > Random 100 c4 c5 c6;
SUBC>   Uniform 0.0 1.0.

MTB > Describe C4 C5 C6;
SUBC>   Mean;
SUBC>   SEMean;
SUBC>   StDeviation;
SUBC>   QOne;
SUBC>   Median;
SUBC>   QThree;
SUBC>   Minimum;
SUBC>   Maximum;
SUBC>   Skewness;
SUBC>   Kurtosis;
SUBC>   N.
Descriptive Statistics: C4, C5, C6 

Variable    N    Mean  SE Mean   StDev  Minimum      Q1  Median      Q3  Maximum  Skewness
C4        100  0.5171   0.0299  0.2987   0.0245  0.2395  0.5329  0.7387   0.9945     -0.01
C5        100  0.4610   0.0306  0.3059   0.0003  0.1899  0.4287  0.7337   0.9905      0.16
C6        100  0.5463   0.0269  0.2691   0.0443  0.3355  0.5581  0.7844   0.9971     -0.17

Variable  Kurtosis
C4           -1.19
C5           -1.26
C6           -1.11

MTB > GSummary  C4 C5 C6.
Summary Report for C4 
Summary Report for C5 
Summary Report for C6 

MTB > PPlot C4 C5 C6;
SUBC>   Normal;
SUBC>   Symbol;
SUBC>   FitD;
SUBC>   Grid 2;
SUBC>   Grid 1;
SUBC>   MGrid 1;
SUBC>   Panel.
Probability Plot of C4, C5, C6 


Comments:
---------
All 3 columns generated are fitted poorly by the normal distribution.

The histograms lack tails (going down like in a normal distribution),
and the kurtosis is negative and quite strong for all distributions. The
normal plots all have the same characteristic shape with curves that
bend downwards to the left and upwards to the right - these patterns
reflect the too short left and right tails of the distribution.