Supplementary Exercise 1.145 of IPS7e
-------------------------------------

Minitab commands and output:

MTB > WOpen "H:\VHM\VHM801\Datasets\Minitab\Chapter 1\ex01_145.mtw".
Retrieving worksheet from file: 'H:\VHM\VHM801\Datasets\Minitab\Chapter
1\ex01_145.mtw'
Worksheet was saved on 20/09/2014

MTB > Describe 'satm' 'gpa';
SUBC>   By 'sex';
SUBC>   Mean;
SUBC>   SEMean;
SUBC>   StDeviation;
SUBC>   QOne;
SUBC>   Median;
SUBC>   QThree;
SUBC>   Minimum;
SUBC>   Maximum;
SUBC>   Skewness;
SUBC>   Kurtosis;
SUBC>   N;
SUBC>   NMissing;
SUBC>   GBoxplot.
Descriptive Statistics: satm, gpa 

Variable  sex    N  N*    Mean  SE Mean   StDev  Minimum      Q1  Median      Q3  Maximum
satm      1    145   0  611.77     6.98   84.02   400.00  550.00  620.00  670.00   800.00
          2     79   0  565.03     9.33   82.93   300.00  510.00  570.00  630.00   740.00

gpa       1    145   0  2.6077   0.0670  0.8068   0.1200  2.1350  2.7500  3.1900   4.0000
          2     79   0  2.6857   0.0820  0.7288   0.3900  2.2500  2.7200  3.3300   4.0000

Variable  sex  Skewness  Kurtosis
satm      1       -0.06     -0.48
          2       -0.47      0.55

gpa       1       -0.71      0.35
          2       -0.59      0.31

Boxplot of satm 
Boxplot of gpa 

Comments:
---------
The boxplot of SAT math scores show distributions of similar shape, but
the distribution for men (sex=1) appears to be centred at a higher value
than for women. The difference in means is approximately 50 units. Both
distributions appear fairly symmetric, and their spread is almost the
same (when measured by the standard deviation).

In contrast, the two distributions of gpa values appear to be centred at
approximately the same value (around 2.7), but both distributions appear
a bit left-skewed with longer left than right tails, with a more pronounced
left-skewness for the men. 

Next, we produce the probability plots. It is probably easier to look at
the plots in separate panels for men and women. This is most easily
achieved in Minitab by choosing Probability Plot-Single, and adding
the panels under the Multiple Graphs-By Variables tab.

MTB > PPlot 'satm' 'gpa';
SUBC>   Normal;
SUBC>   Symbol;
SUBC>   FitD;
SUBC>   Grid 2;
SUBC>   Grid 1;
SUBC>   MGrid 1;
SUBC>   Panel 'sex'.
Probability Plot of satm 
Probability Plot of gpa 

Comments:
---------
The normal probability plots for satm are nicely straight, and the
normality tests are clearly non-significant (P>>0.05), leading us to the
conclusion that there these values can be viewed as approximately
normally distributed. The boxplot for women shows one low suspected
outlier, and this point is also outside the normal probability limits,
but overall the distribution (with this observation included) seems well
approximated by a normal distribution.

The normal distributio plots for gpa look less straight because of some
upwards curvature at the lower end, reflecting the left skewness. The
normality test is clearly significant (P<0.005) for the men and non-significant
for the women. The distributions do not appear that different, so the
difference in conclusion from the normality test may be mostly a
reflection of the different sample sizes (almost twice as many men as
women). With a larger sample size, any deviations from a normal
distribution are less likely to appear as the result of random
fluctuations. In conclusion, we should be cautious in assuming normal
distributions for both the gpa distributions.

In response to the question asked in the exercise, ignoring outliers
does not seem to substantially alter our impression of the
distributions. Generally speaking it is not recommended to "ignore" outliers
for the sole purpose of achieving distributions that seem closer to
normal. Later in the course we will discuss how to deal with non-normal 
distributions in the context of statistical inference.