Two objects with unknown weights ₁ and ₂ are weighed separately on a scale and together (both placed on the scale), giving three measurements x1, x2 and x3. It is known from previous experience with the scale that measurements are independent and normally distributed about the true weights with variance 1.

We have previously shown that the MLES are 1 = (2x₁ + x₃ x₂)/3 and 2 = (2x₂+x₃x₁)/3 and the log likelihood is l(1, 2) = (x₁₁)²/2(x₂₂)²/2(x₃₁₂)²/2. Consider the parameterization = (₁, )^T where = _{2 1} and the data x₁ = 15.6, x₂ = 29.3 and x₃ = 45.8.

a) Obtain the MLE of

b) Obtain the information matrix I(1, )

c) Obtain an approximate 95% confidence interval for