Minimization of the variance of an estimator.

Question

I have the following problem,

In the design of an experiment to compare two treatments using independent samples, the budget reaches a total of 20 observations

a)How to assign the observations to the samples so as to minimize the variance of the estimator $X$, the difference of means? Both samples have the same variance.

b)How to assign the observations in order to maximize the power of the test for $X$?

Can give me some hint, it's the first time I see a problem like this. Thanks!!

Answer 1

Step 1: Let the sample sizes be $n_1$ and $n_2$, and the variances be $\sigma^2_1$ and $\sigma^2_2$.

Step 2: Write down the variance of the estimator, $Var(X)$, in terms of the above values.

Step 3: Note that $n_1 + n_2 = 20$.

Step 4: Minimise $Var(X)$ as a function of $n_1$.