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.

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.

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