The question asks for the conditions under which the variance of a sample mean from Simple Random Sampling With Replacement (SRSWR) is equal to the variance of a sample mean from stratified sampling with proportional allocation.
Variance of SRSWR Sample Mean (T1)
For a sample mean $T_1$ obtained through SRSWR from a population with variance $\sigma^2$, the variance is given by:
where $n$ is the sample size.
Variance of Stratified Sample Mean (T2)
For a stratified sample mean $T_2$ drawn using proportional allocation from $L=4$ strata, the variance is:
where $W_h = N_h/N$ is the stratum weight, $N_h$ is the size of stratum $h$, $N$ is the total population size, $\sigma_h^2$ is the variance within stratum $h$, and $n_h$ is the sample size from stratum $h$.
With proportional allocation, $n_h = n W_h$. Substituting this into the variance formula gives:
Condition for Variance Equality
We need to find conditions such that $\text{Var}(T_1) = \text{Var}(T_2)$. Equating the two expressions:
This simplifies to:
Decomposition of Population Variance
The overall population variance $\sigma^2$ can be expressed in terms of within-stratum variances and between-stratum variance:
Here, $\mu$ is the population mean, $\mu_h$ is the mean of stratum $h$, and $\sum_{h=1}^4 W_h (\mu_h - \mu)^2$ is the between-stratum variance component (often denoted as $B^2$).
Identifying the Sufficient Condition
For the equality $\sigma^2 = \sum_{h=1}^4 W_h \sigma_h^2$ to hold, the between-stratum variance component must be zero:
Since $W_h > 0$ (strata have positive weights), this condition implies that $(\mu_h - \mu)^2 = 0$ for all $h=1, 2, 3, 4$. This means:
This result indicates that the mean of each stratum must be equal to the overall population mean. This is precisely the condition stated in Option 3.
Conclusion
Therefore, the condition that strata means are same ($\mu_h = \mu$ for all $h$) is sufficient for $\text{Var}(T_1) = \text{Var}(T_2)$.
Important Questions from Ratio And Regression
- Consider the problem of drawing a sample of size 2 from a finite population of size 20. The sampling is done with replacement using probability proportional to size sampling scheme. The normed size measures $p_1, \cdots, p_{20}$ are given by $p_i = \frac{1}{40}$, $i = 1, \cdots, 10, \; p_i = \frac{3}{40}$, $i = 11, \cdots, 20$. The expected number of distinct units drawn is
Suppose there are $k$ strata of $N = kM$ units each with size $M$. Draw a sample of size $n_i$ with replacement from the $i^{\text{th}}$ stratum and denote by $\bar{y}_i$ the sample mean of the study variable selected in the $i^{\text{th}}$ stratum, $i = 1, 2, \dots, k$. Define
$$ \bar{y}_s = \frac{1}{k}\sum_{i=1}^k \bar{y}_i \text{ and } \bar{y}_w = \frac{\sum_{i=1}^k n_i \bar{y}_i}{n} $$
Which of the following is necessarily true?- Suppose we draw a random sample of size $n$ from a population of size $N$, where $1 < n < N$, using simple random sampling without replacement scheme. Let $P$ be the population proportion of units possessing a particular attribute and $p$ be the corresponding sample proportion. Which of the following is an unbiased estimator for $P(1 - P)$?
Suppose there are $k$ groups each consisting of $N$ boys. We want to estimate the mean age $\mu$ of these $kN$ boys. Fix $1 < n < N$ and consider the following two sampling schemes.
I. Draw a simple random sample without replacement of size $kn$ out of all $kN$ boys.
II. From each of the $k$ groups draw a simple random sample with replacement of size $n$.
Let $\bar{Y}$ and $\bar{Y}_G$ be the respective sample mean ages for the two schemes. Which of the following are true?
- A sample of size $n (\ge 2)$ is drawn without replacement from a finite population of size $N$, using an arbitrary sampling scheme. Let $\pi_i$ denote the inclusion probability of the $i$-th unit and $\pi_{ij}$, the joint inclusion probability of units $i$ and $j, 1 \le i < j \le N$. Which of the following statements is always true?
