Supplementary Exercise 7.143 of IPS7e
-------------------------------------

Data: OC (osteocalcin) measurements (in mg/ml blood) for 31 healthy women
between 11 and 32 years.

Model: the 31 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.

Minitab commands and output:

MTB > WOpen "H:\VHM\VHM801\Datasets\Minitab\Chapter 7\ex07_143.mtw".
Retrieving worksheet from file: 'H:\VHM\VHM801\Datasets\Minitab\Chapter
7\ex07_143.mtw'
Worksheet was saved on 07/10/2014

MTB > Describe  'oc';
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: oc 

Variable   N  N*   Mean  SE Mean  StDev  Minimum     Q1  Median     Q3  Maximum
oc        31   0  33.42     3.52  19.61     8.10  17.90   30.20  47.70    77.90

Variable  Skewness  Kurtosis
oc            0.84     -0.19

Boxplot of oc

MTB > Stem-and-Leaf 'oc';
SUBC>   Trim.
Stem-and-Leaf Display: oc 

Stem-and-leaf of oc  N  = 31
Leaf Unit = 1.0

 2   0  89
 3   1  0
 10  1  5677799
 15  2  00004
 15  2
(3)  3  011
 13  3  568
 10  4  04
 8   4  7
 7   5  244
 4   5  6
 3   6
 3   6  8
 2   7
 2   7  67

MTB > GSummary  'oc'.
Summary Report for oc 

MTB > PPlot 'oc';
SUBC>   Normal;
SUBC>   Symbol;
SUBC>   FitD;
SUBC>   Grid 2;
SUBC>   Grid 1;
SUBC>   MGrid 1.
Probability Plot of oc 
The P-value of the Anderson-Darling test of normality is 0.009

MTB > Onet 'oc'';
SUBC>   Confidence 95.0;
SUBC>   Alternative 0.
One-Sample T: oc 

Variable   N   Mean  StDev  SE Mean      95% CI
oc        31  33.42  19.61     3.52  (26.22, 40.61)

Answers to questions:
---------------------
(a)
The stemplot, histogram and boxplot all show that the data are 
right-skewed and not approximated well by a normal distribution. The test
(Anderson-Darling) for normality is clearly significant. The skewness is
0.84. The distribution appears to be unimodal. There are no obvious
outliers, and the 1.5*IQR rule does not flag any observations as suspected
outliers. Beware that for skewed data this rule often flags some observations
in the heavier tail (the right tail for a right-skewed distribution), even
if these are not outlying at all. This is because the rule is based on a
symmetrical distribution.

(b)
The confidence interval is given above. It is APPROXIMATE only, because 
the observations themselves are clearly not normally distributed (but the sample
mean is nevertheless approximately normally distributed, by the CLT). It is not
clear whether this is a situation where the guidelines (7L-3) for use of the 
t-procedure are met: the skewness is quite pronounced, and the number of 
observations is less than 40.