Two objects of unknown weights w1 and w2 are weighed on an error-prone pan balance in the following way: (1) object 1 is weighed by itself, and the measurement is 3 g; (2) object 2 is weighed by itself, and the result is 3 g; (3) the difference of the weights (the weight of object 1 minus the weight of object 2) is measured by placing the objects in different pans, and the result is 1 g; (4) the sum of the weights is measured as 7g. The problem is to estimate the true weights of the objects from these measurements.
a. Set up a linear model, Y = Xβ + e. (Hint: The entries of X are 0 and ±1.)
b. Find the least squares estimates of w1 and w2.
c. Find the estimate of σ2.
d. Find the estimated standard errors of the least squares estimates of part (b).
e. Estimate w1 − w2 and its standard error.
f. Test the null hypothesis H0: w1 = w2.
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