Math 6010 • Linear Models

A scale has two pans. The measurement given by the scale is the
difference between the weights on pan 1 and pan 2 plus a random error. Thus, if
a weight \(\mu_1\) is put on pan 1 and a weight \(\mu_2\) is put on pan 2, then the
measurement is \(Y=\mu_1-\mu_2+\epsilon\). Suppose that \(\E[\epsilon]=0\),
\(\Var(\epsilon)=\sigma^2\), and in repeated use of the scale observations \(Y_i\)
are independent.

Suppose two objects, \#1 and \#2, have weights \(\beta_1\) and \(\beta_2\).
Measurements are then taken as follows:

\begin{itemize}
\item Object \#1 is put on pan 1 and nothing on pan 2
\item Object \#2 is put on pan 2 and nothing on pan 1
\item Object \#1 is put on pan 1 and object \#2 is put on pan 2
\item Objects \#1 and \#2 are both put on pan 1 and nothing on pan 2
\end{itemize}

\textbf{(a)} Let \(\mathbf{Y}=(Y_1,Y_2,Y_3,Y_4)^\prime\) be the vector of
observations. Formulate this as a linear model.

\textbf{(b)} Find the least squares estimators of \(\beta_1\) and \(\beta_2\).

\textbf{(c)} Find the variances of the least squares estimators as well as the
covariance between them.