Typically, in reporting experimental data, at any given
i
, the mean
i
and standard deviation
i
are given. Examples are shown in Figures 5.1 and 8.11. The objective of this exercise is to derive a regression procedure that produces a predicted value
that: i) of foremost importance, falls within the interval
of secondary importance, has
i
.
(a) Setting
on the same axes, sketch
i
as a function of
for
= 2, 4, 8. Use this to explain why taking a larger
has the effect of giving more weight to objective (i).
(b) Does
i
satisfy the three expected properties of an error function given in Section 8.2.2?
(c) Taking
= 2,
=
+
, and the error function
find
and
that minimize
.
(d) Suppose one gets better, and presumably more expensive, experimental equipment so the standard deviations are all reduced by a factor of, say, 10. Assuming the same
i
values are obtained, explain why the answer in part (c) is unchanged.