Suppose we have an ordinary household scale such as might be used in a kitchen. When an object is placed on the scale, the reading is a combination of the true weight plus a random error. You have two coins of unknown values B1 and B2. To estimate the weights of the coins, you take four observations:
- Put coin 1 on the scale and observe y1
- Put coin 2 on the scale and observe y2
- Put both coins on the scale and y3
- Put both coins on the scale and observe y4
(a) Write a linear model in matrix form and find the least-squares estimate of the coins using the usual formula (X'X)^-1 X'y.
Regarding the question from letter (a), my understanding is that for either B1 or B2, they can be estimated by averaging the weights (for example, for B1 it would be adding the weights from y1, y3 and y4). Is this assumption valid?
I thought it was, and so I tried to set up the formula for B1:
B1 = 1/3*K1*Y1 + 0*K2*Y2 + 1/6*K3*Y3 + 1/6*K4*Y4 + e1 + e2 + e3 + e4, and B2 = 0*K1*Y1 + 1/3*K2*Y2 + 1/6*K3*Y3 + 1/6*K4*Y4 _ e1 + e2 + e3 + e4,
Kiis the slope parameter estimate for the
eiis the residual for the
ithcoin. The numbers 1/3 and 1/6 (1/3 * 1/2) to find the average weight of
B1. Am I totally off base with this reasoning and subsequent setup of my linear model?
Can I also assume that
y3is found by simply adding
y2? I didn't think I could, because the readings would probably be different and thus have different residuals.
yin the formula correspond with the estimated column vector containing predicted