tratified Sampling Variance Explanation

This solution explains the formula for the variance of the stratified sample mean, denoted as $Var(\bar{y}_{stratified})$, under the specific conditions provided in the question.

Key Variables and Definitions

To understand the formula, let's first define the variables used:

• $k$: Represents the total number of strata within the population.
• $N_i$: Denotes the number of sampling units belonging to the $i^{th}$ stratum in the population.
• $N$: Represents the total number of sampling units in the entire population, calculated as the sum of units across all strata ($N = \sum_{i=1}^{k} N_i$).
• $n_i$: Indicates the number of sampling units selected from the $i^{th}$ stratum to be included in the sample.
• $n$: Represents the total sample size, which is the sum of sample units from all strata ($n = \sum_{i=1}^{k} n_i$).
• $P_i$: Represents the proportion of the total population units that are in the $i^{th}$ stratum. It is calculated as $P_i = \frac{N_i}{N}$.
• $S_i^2$: Denotes the variance calculated for the variable of interest within the $i^{th}$ stratum of the population.

Understanding Proportional Allocation

The question specifies that $n_i \propto N_i$. This proportionality means that the sample size chosen for each stratum ($n_i$) is directly related to the size of that stratum in the population ($N_i$). This specific method is called proportional allocation. With proportional allocation, the proportion of the sample taken from each stratum is the same as the proportion of the population in that stratum:

$ \frac{n_i}{n} = \frac{N_i}{N} $

Using the definition of $P_i$, this relationship can also be expressed as:

$ n_i = n \frac{N_i}{N} = n P_i $

Formula for Stratified Sample Mean Variance

Considering the provided options and the indicated correct answer, the variance of the stratified sample mean ($\bar{y}_{stratified}$) under the condition $n_i \propto N_i$ is given by the following formula:

$ Var(\bar{y}_{stratified}) = \left(\frac{1}{N}-\frac{1}{n}\right)\sum_{i=1}^{k}P_i^2 S_i^2 $

Let's analyze the components of this formula:

• The Term $\left(\frac{1}{N}-\frac{1}{n}\right)$
  • This component is related to the Finite Population Correction (FPC). FPC factors are used in sampling without replacement from finite populations to adjust the variance estimate. Typically, FPC terms are positive when the sample size $n$ is less than the population size $N$. The specific form $\left(\frac{1}{N}-\frac{1}{n}\right)$ would result in a negative value if $n < N$. Standard statistical formulas often use related forms like $\left(\frac{1}{n}-\frac{1}{N}\right)$ or $\left(1 - \frac{n}{N}\right)$. However, based on the provided correct option, this specific structure is used here.
• The Term $\sum_{i=1}^{k}P_i^2 S_i^2$
  • This part involves summing contributions from each stratum, indexed from $i=1$ to $k$.
  • Within the sum, for each stratum $i$, it calculates the product of the square of the stratum's population proportion ($P_i^2$) and the square of the stratum's variance ($S_i^2$).
  • It is worth noting that common formulas for stratified variance typically involve terms like $\sum P_i S_i^2$ or $\sum P_i S_i^2 (1-P_i)$. The inclusion of squared proportions ($P_i^2$) and squared variances ($S_i^2$) makes this summation term different from the standard formulations.

In summary, the variance of the stratified sample mean ($\bar{y}_{stratified}$), when employing proportional allocation ($n_i \propto N_i$), is determined by the formula $\left(\frac{1}{N}-\frac{1}{n}\right)\sum_{i=1}^{k}P_i^2 S_i^2$, according to the structure presented in the question's options.