Extra exercise 18
-----------------
(continuation of previous supplementary exercises 7.102-104 with data for added groups)

Minitab commands, with comments below

MTB > WOpen "H:\VHM\VHM801\Datasets\Minitab\Chapter 7\ex07_102.mtw".
Retrieving worksheet from file: 'H:\VHM\VHM801\Datasets\Minitab\Chapter
7\ex07_102.mtw'
Worksheet was saved on 02/11/2014

MTB > Describe 'r';
SUBC>   By 'group2';
SUBC>   Mean;
SUBC>   SEMean;
SUBC>   StDeviation;
SUBC>   QOne;
SUBC>   Median;
SUBC>   QThree;
SUBC>   Minimum;
SUBC>   Maximum;
SUBC>   Skewness;
SUBC>   Kurtosis;
SUBC>   N.
Descriptive Statistics: r 

Variable  group2     N    Mean  SE Mean  StDev  Minimum      Q1  Median     Q3  Maximum
r         computer  20   0.450    0.495  2.212   -3.000  -1.000   0.500  2.000    4.000
          none      14   0.786    0.853  3.191   -6.000  -1.000   0.000  2.500    7.000
          piano     34   3.618    0.524  3.055   -3.000   2.000   4.000  6.000    9.000
          singing   10  -0.300    0.473  1.494   -4.000  -1.000   0.000  1.000    1.000

Variable  group2    Skewness  Kurtosis
r         computer     -0.12     -0.93
          none          0.01      1.02
          piano        -0.36     -0.28
          singing      -1.86      4.26

MTB > Dotplot ( 'r' ) * 'group2'.
Dotplot of r 

MTB > PPlot 'r';
SUBC>   Normal;
SUBC>   Symbol;
SUBC>   FitD;
SUBC>   Grid 2;
SUBC>   Grid 1;
SUBC>   MGrid 1;
SUBC>   Panel 'group2'.
Probability Plot of r 

The P-values for the A-D tests for normality are:
  Computer - P=0.32
  None - P=0.31
  Piano - P=0.23
  Singing - P=0.015

Comments:
---------
The descriptive statistics show the 4 groups to differ in several ways. Looking
at the mean, the piano groups is clearly highest. Looking at the standard
deviation, the singing group is clearly lowest, and the IPS guideline is
violated. Also, testing homogeneity of variances does not give clear
significance with any of the two tests (see below). Therefore it might be ok to
use the variance homogeneity assumption, with a small reservation perhaps.
As to the normal distribution, only the singing group presents some problems (P=0.015).
The low value of -4 falls outside the others (and if it was removed, the standard
deviation would drop dramatically). It is fair to say that we may proceed with the
analysis, with a small reservation for the singing group. If results are not
clear, we may want to rerun the analysis without that group.

Statistical model: X_ij is normally distributed N(mu_i,sigma), and all obs. are
independent.

MTB > VarTest 'r' 'group2';
SUBC>  Confidence 95.0; 
SUBC>  GInterval;
SUBC>   NoDefault; 
SUBC>  TMethod; 
SUBC>  TBonferroni; 
SUBC>  TTest.
Test for Equal Variances: r versus group2 

Method
Null hypothesis         All variances are equal
Alternative hypothesis  At least one variance is different
Significance level      a = 0.05

95% Bonferroni Confidence Intervals for Standard Deviations

  group2   N    StDev          CI
computer  20  2.21181  (1.66370, 3.36012)
    none  14  3.19082  (1.80321, 6.87230)
   piano  34  3.05520  (2.32745, 4.32846)
 singing  10  1.49443  (0.45553, 6.53495)

Individual confidence level = 98.75%

Tests
                           Test
Method                Statistic  P-Value
Multiple comparisons          -    0.130
Levene                     1.73    0.169
 
Test for Equal Variances: r vs group2 
 
MTB > OneWay;
SUBC>   Response 'r';
SUBC>   Categorical 'group2';
SUBC>   IType 0;
SUBC>   GMCI;
SUBC>   TMTest;
SUBC>   GIntPlot;
SUBC>   TMethod;
SUBC>   TFactor;
SUBC>   TANOVA;
SUBC>   TSummary;
SUBC>   TMeans;
SUBC>   Nodefault.
One-way ANOVA: r versus group2 

Method
Null hypothesis         All means are equal
Alternative hypothesis  At least one mean is different
Significance level      a = 0.05
Equal variances were assumed for the analysis.

Factor Information
Factor  Levels  Values
group2       4  computer, none, piano, singing

Analysis of Variance

Source  DF  Adj SS  Adj MS  F-Value  P-Value
group2   3   207.3  69.094     9.24    0.000
Error   74   553.4   7.479
Total   77   760.7

Model Summary
      S    R-sq  R-sq(adj)  R-sq(pred)
2.73475  27.25%     24.30%      20.04%

Means
group2     N    Mean  StDev       95% CI
computer  20   0.450  2.212  (-0.768, 1.668)
none      14   0.786  3.191  (-0.671, 2.242)
piano     34   3.618  3.055  ( 2.683, 4.552)
singing   10  -0.300  1.494  (-2.023, 1.423)

Pooled StDev = 2.73475
Interval Plot of r vs group2 

Comments:
---------
Summary of statistical testing:
  H0: mu1=mu2=mu3=mu4
  Ha: some of the mu's differ
  Test statistic: F=9.24, degrees of freedom =(3,74)
  P-value: < 0.0005 (from table above)
Conclusion: test strongly significant => some differences exist between groups,
that is, the type of lesson has some impact on test score changes.

The estimated means and confidence intervals strongly suggest that the 
piano group has higher score changes than the other groups, which have more or less
equal score changes.

Added Minitab listing for multiple comparisons below; among 4 groups there are 4*3/2=6 comparisons, 
so the individual error rate is given as 0.05/6 = 0.0083333.

MTB > OneWay;
SUBC>   Response 'r';
SUBC>   Categorical 'group2';
SUBC>   IType 0;
SUBC>   Fisher 0.0083333;
SUBC>   GMCI;
SUBC>   TGrouping;
SUBC>   TMTest;
SUBC>   GIntPlot;
SUBC>   TMethod;
SUBC>   TFactor;
SUBC>   TANOVA;
SUBC>   TSummary;
SUBC>   TMeans;
SUBC>   Nodefault.
One-way ANOVA: r versus group2 
...
Fisher Pairwise Comparisons 

Grouping Information Using the Fisher LSD Method and 99.1667% Confidence

group2     N    Mean  Grouping
piano     34   3.618  A
none      14   0.786    B
computer  20   0.450    B
singing   10  -0.300    B

Means that do not share a letter are significantly different.

Fisher Individual Tests for Differences of Means
                      Difference       SE of                             Adjusted
Difference of Levels    of Means  Difference     99.1667% CI    T-Value   P-Value
none - computer            0.336       0.953  (-2.248,  2.919)     0.35     0.726
piano - computer           3.168       0.771  ( 1.078,  5.257)     4.11     0.000
singing - computer         -0.75        1.06  ( -3.62,   2.12)    -0.71     0.481
piano - none               2.832       0.868  ( 0.478,  5.186)     3.26     0.002
singing - none             -1.09        1.13  ( -4.16,   1.98)    -0.96     0.341
singing - piano           -3.918       0.984  (-6.585, -1.251)    -3.98     0.000

Simultaneous confidence level = 95.93%
Fisher Individual 99.1667% CIs 
Interval Plot of r vs group2 

Comments:
---------
None of the confidence intervals for comparisons with the piano group 
contain zero, and significant differences at the overall 5% error level
therefore exist between the piano and all other groups. The 3 remaining
confidence intervals all contain zero; there are no significant differences between the
other groups. Therefore, we conclude that there is evidence for the piano group 
to be above all others, but there is no evidence to distinguish the other groups.
This conclusion is represented by the letter-coding graphica before the comparisons 
by the confidence intervals: the Piano group has letter A, and all the other groups 
have letter B.