I am begginer in sampling, so I need help with this exercise.
I am given population of size $N$. For every item $k$ in population we given two data $(Y_k, X_k), Y_k \in \{a,b,c,d\}, X_k \in \mathbb{R}$. We divide population in four stratums (based on $Y_k$ of each item). Stratums have sizes $N_1,N_2,N_3,N_4$.
We do stratified random sampling (sample size $n_i$ for stratum i), and have to calculate sample mean and population variance of each stratum and between stratums.
Procedure for sample mean:
- We calculate the proportions of stratums: $w_i = N_i/N$
- Do simple random sampling for each stratum and calculate stratum sample mean $\bar{X_i}$.
- Sample mean is: $\bar{X} = \sum w_i \bar{X_i}$.
Equation for population variance inside stratum 1 (symetrical for others): $\frac{N_1-n_1}{(N_1-1)n_1} \frac{\sum_{k=1}^{N_1} (X_k - \bar{X_1})^2}{N_1}$.
I am lost on how to do population variance between stratums. What does that even mean? Are other equations right? Thank you.