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Unbiased estimator of linear contrast

I'm getting really muddled up with parts bii) and biii)

For the first part, I've written this on my page but not sure if it's going anywhere correct;

L = \sum_{i=1}^k c_ia_i

\hat{L} = \sum_{i=1}^k c_i\hat{a}_i

Unparseable latex formula:

\hat{a}_i = \bar{Y}_i_. - \bar{Y}_._.

Unparseable latex formula:

E(\hat{L}) = E(\sum_{i=1}^k c_i\hat{a}_i) = \sum_{i=1}^k c_iE((\bar{Y}_i_. - \bar{Y}_._.))

Then I'm a bit stuck.

Part biii) (I just want to check this is correct)

Unparseable latex formula:

Var(\bar{Y}_i_.) = Var (\frac{1}{n_i} \sum_{j = 1}^{n_i} Y_i_j)

Unparseable latex formula:

\frac{1}{n_i^2} \sum_{j = 1}^{n_i} Var( \mu + a_i + e_i_j)

=\frac{\sigma^2}{n_i}

Unparseable latex formula:

Var(\sum_{i = 1}^{k}c_i\bar{Y}_i_.) = \sigma^2 \sum_{i = 1}^{k} \frac{c_i^2}{n_i}

In this part, I've left out a lot of the steps I've written on the page to avoid typing it all out in latex but if there's any confusion, I can explain my logic.