A new process for measuring the concentration of a chemical
in water is being investigated. A total of n samples are prepared
in which the concentrations are the known numbers
for
;
the new process is used to measure the concentrations for these samples.
It is thought likely that the concentrations measured by the new process,
which we denote
, will be related to the true concentrations via
![](img18.gif)
where the
are independent, have mean 0 and all have the
same variance
which is unknown.
- If this model is fitted by least squares, (that is by minimizing
) show that the least squares estimate of
is
![](img23.gif)
- Show that the estimator in part (a) is unbiased.
- Compute (give a formula for) the standard error of
.
- The error sum of squares for this model is
which may be shown to have n-1 degrees of freedom.
If the
are the numbers 1, 2, 3 and 4,
and the
error sum of squares is 0.12 find a 95% confidence interval for
and explain what further assumptions you must make to do so.
- Show that the estimator
![](img29.gif)
is also unbiased.
- Compute (give a formula for) the standard error of
. Which
is bigger, the standard error of
or that of
?
- Show that the mle of
in this model is
, the least
squares estimate, if the
have normal distributions.