STA 437/1005, Assignment #2, Question #2.

This discussion refers to plots and R output on the course web page.

The pairwise plots of differences show no clear indications of
non-normality or of outliers.  Due to the very small sample size, we
could hope to detect non-normality only if it were extreme, so a fair
degree of non-normality is still possible, but it makes sense to
proceed on the assumption that the differences are normally
distributed.  Only one outlier, with weight difference over 20
kilograms, is apparent.  Common sense says that such a difference can
arise from different diet, even if the genes are the same, so there is
no reason to think that this observation is erroneous.  I have
retained it in the data.

The T-squared test of the null hypothesis of the mean difference being
zero for all five variables produces a p-value of 0.54036, which
indicates no evidence that the null hypothesis is false.  So the
researcher's idea that there might be a difference has not been
confirmed.

The 90% confidence intervals from the T-squared test all contain zero
(consistent with the p-value from the hypothesis test).  The
univariate 90% confidence intervals based on the t statistic also all
contain zero, except for the difference in area, where the interval is
(3.757, 134.965).  However, when the Bonferroni correction is applied
in order to produce simultaneous 90% confidence intervals, all
intervals, including that for area, contain zero.  Unless the
researcher had some reason to think (even before looking at the data)
that the area variable was particularly likely to show a non-zero
difference in mean, the univariate confidence interval for the
difference in area should be discounted, since even if the null
hypothesis is true, it is fairly likely that at least one of five 90%
confidence intervals would not contain zero.

The sample size for this study is very small, so only quite large
differences in mean between the first-born and second-born twin would
be detectable.  If the researcher thinks has good reason to expect a
difference (one within the bounds of the confidence intervals obtained
from this study), they should try to obtain more data, enough so that
differences of moderate size would be detectable.