I am trying to figure out what sample size recommendations to make in this case when the samples have different sizes. Since their sizes are different the variance of each will be different.
Suppose the dataset is to be used to inform the design of a new study in the future. Assume $\alpha=0.05$ and $power=0.80$ although the researcher is not sure what the smallest scientifically significant mean difference between the two groups should be. What recommendation would you make about the sample sizes $n_1$ and $n_2$ (including the relationship between $n_1$ and $n_2$ and the magnitude of $n_1,n_2$)? Provide reasoning.
healthy <- c(1870,1324,1446,1325,1759,1652,1364,1515,1065)
infected <- c(1121,408,184,16,741,170,991,711,734,202,893,742,335,444)
Update
So far the recommendation I have made is that we should try to make the sample sizes equal to avoid issues that could occur when the variances are unequal.
Previously, using this data a two-sample independence t-test was conducted. So we assumed that the two samples were independent and that the variances were equal.
If we made this assumption again:
When $\sigma_1^2 = \sigma_2^{2}$:
$S^2_p = \frac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-1}$
$E[S^2_p] = \sigma^2$
When $\sigma_1^2 \neq \sigma_2^{2}$:
$E[S^2_p] = \frac{(n_1-1)\sigma_1^2+(n_2-1)\sigma_2^2}{n_1+n_2-1}$
$\hat{var(\bar{Y_1} - \bar{Y_2})} = (\frac{1}{n_1} + \frac{1}{n_2} )S_p^2$
$E[\hat{var(\bar{Y_1} - \bar{Y_2})] = Var(\bar{Y_1} - \bar{Y_2})} $
Which may cause issues due to their sample sizes being different.
I am not sure if this is enough to show this.
Regarding the magnitude of $n_1$ and $n_2$, I am not sure how to show this. Would appreciate a hint.
