# Linear Models-Given three objects of unknown weight and a set of scales with one pan.

#### AMcC

##### New Member
Hi all,

I am new to this and pure desperation is starting to set in. Below is my problem:

Three objects with unknown weights are weighed on a set of scales in all possible combinations giving seven independent measurements ($$y_1,...,y_7$$). Each measurement includes a random error term, , with $$E[\epsilon]=0$$and $$Var(\beta)=\sigma^2$$

(a) Set up a linear model with a parameter to represent the weight of each of the three objects.
(b) Find the least squares estimates of the weights.
(c) Find the var($$\hat{\beta}$$).
(d) Estimate the total weight of the three objects and give an expression for the variance of this total.

I think that
$$y_1=\beta_1 + \epsilon_1$$
$$y_2=\beta_2 +\epsilon_2$$
$$y_3=\beta_3 + \epsilon_3$$
$$y_4=\beta_1 +\beta_2 + \epsilon_4$$
$$y_5=\beta_1 +\beta_3 + \epsilon_5$$
$$y_6=\beta_2 +\beta_3 + \epsilon_6$$
$$y_7=\beta_1 +\beta_2 + \beta_3 + \epsilon_7$$

Then I calculated my Y matrix to be a 1*7 with entries $$y_1...y_7$$

And my X matrix to be
1 0 0
0 1 0
0 0 1
1 1 0
1 0 1
0 1 1
1 1 1

This is where I am uncertain. Since there is only one pan, should I have a 7*3 matrix or should it be a 7*1? In which case I get a little lost.

I would appreciate any help. Thank you.

Last edited:

#### AMcC

##### New Member
Re: Linear Models-Given three objects of unknown weight and a set of scales with one

It is the matrix that I have posted, just incase someone in the future comes across this thread