A chemist has seven light objects to weigh on a balance pan scale. The standard deviation of each weighing is denoted by $\sigma$. In a 1935 paper, Frank Yates [Yates, 1935] suggested an improved technique by weighing all seven objects together, and also weighing them in groups of three. The groups are chosen so that each object is weighed four times altogether, twice with any other object and twice without it.

Let $y_{1}, \ldots, y_{8}$ be the readings from the scale so that the equations for determining the unknown weights, $\beta_{1}, \ldots, \beta_{7}$, are

\[\begin{aligned}& y_{1}=\beta_{1}+\beta_{2}+\beta_{3}+\beta_{4}+\beta_{5}+\beta_{6}+\beta_{7}+\epsilon_{1} \\& y_{2}=\beta_{1}+\beta_{2}+\beta_{3}+\epsilon_{2} \\& y_{3}=\beta_{1}+\beta_{4}+\beta_{5}+\epsilon_{3} \\& y_{4}=\beta_{1}+\beta_{6}+\beta_{7}+\epsilon_{4} \\& y_{5}=\beta_{2}+\beta_{4}+\beta_{6}+\epsilon_{5} \\& y_{6}=\beta_{2}+\beta_{5}+\beta_{7}+\epsilon_{6} \\& y_{7}=\beta_{3}+\beta_{4}+\beta_{7}+\epsilon_{7} \\& y_{8}=\beta_{3}+\beta_{5}+\beta_{6}+\epsilon_{8},\end{aligned}\]

where the $\epsilon_{i}, i=1, \ldots, 8$ are independent errors.

Hotelling [1944] suggested modifying Yates' procedure by placing in the other pan of the scale those of the objects not included in one of his weighings. In other words if the first three objects are to be weighed, then the remaining four objects would be placed in the opposite pan.

a. Write Yates' procedure in matrix form and find the least squares estimates of $\beta$.b. Write Hotelling's procedure in matrix form $\mathbf{y}=X \beta+\epsilon$, where $\mathbf{y}^{\prime}=\left(y_{1}, \ldots, y_{8}\right)$, $\beta^{\prime}=\left(\beta_{1}, \ldots, \beta_{7}\right), \epsilon^{\prime}=\left(\epsilon_{1}, \ldots, \epsilon_{8}\right)$, and $X$ is an $8 \times 7$ matrix. Find the least squares estimate of $\beta$.

c. Find the variance of a weight using Yates' and Hotelling's procedures.

d. If the chemist wanted estimates of the weights with the highest precision, then which procedure (Yates or Hotelling) would you recommend that the chemist use to weigh objects? Explain your reasoning.