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?
$\bar{y}_s$ is unbiased but $\bar{y}_w$ is not unbiased for the population mean.
Stratified Sampling Unbiasedness Check
The problem involves stratified sampling with replacement. We have $k$ strata, each of equal size $M$, making the total population size $N = kM$. The population mean of stratum $i$ is $\bar{Y}_i$. Since strata sizes are equal, the overall population mean is $\bar{Y} = \frac{1}{k}\sum_{i=1}^k \bar{Y}_i$. We analyze the unbiasedness of two estimators, $\bar{y}_s$ and $\bar{y}_w$.
Analysis of Estimator \(\bar{y}_s\)
The estimator $\bar{y}_s$ is the simple average of the sample means from each stratum:
\[ \bar{y}_s = \frac{1}{k}\sum_{i=1}^k \bar{y}_i \]Its expected value is:
\[ E(\bar{y}_s) = E\left(\frac{1}{k}\sum_{i=1}^k \bar{y}_i\right) = \frac{1}{k}\sum_{i=1}^k E(\bar{y}_i) \]As samples are drawn with replacement, the expected value of a sample mean equals the population mean of that stratum, i.e., $E(\bar{y}_i) = \bar{Y}_i$. Thus:
\[ E(\bar{y}_s) = \frac{1}{k}\sum_{i=1}^k \bar{Y}_i \]Since $E(\bar{y}_s)$ equals the population mean $\bar{Y}$, the estimator $\bar{y}_s$ is unbiased.
Analysis of Estimator \(\bar{y}_w\)
The estimator $\bar{y}_w$ weights the sample means by their respective sample sizes $n_i$:
\[ \bar{y}_w = \frac{\sum_{i=1}^k n_i \bar{y}_i}{n} \]where $n = \sum_{i=1}^k n_i$ is the total sample size. Its expected value is:
\[ E(\bar{y}_w) = E\left(\frac{\sum_{i=1}^k n_i \bar{y}_i}{n}\right) = \frac{1}{n}\sum_{i=1}^k n_i E(\bar{y}_i) = \frac{1}{n}\sum_{i=1}^k n_i \bar{Y}_i \]For $\bar{y}_w$ to be unbiased, $E(\bar{y}_w)$ must equal $\bar{Y} = \frac{1}{k}\sum_{i=1}^k \bar{Y}_i$. This requires the condition $\frac{1}{n}\sum_{i=1}^k n_i \bar{Y}_i = \frac{1}{k}\sum_{i=1}^k \bar{Y}_i$ to hold.
Because all strata have equal size ($N_i=M$), this condition is satisfied if and only if the sample sizes are allocated equally ($n_i = n/k$ for all $i$). Since the problem does not state that $n_i$ are equal, $\bar{y}_w$ is not necessarily unbiased.
Conclusion
The estimator $\bar{y}_s$ is necessarily unbiased for the population mean because the strata sizes are equal. The estimator $\bar{y}_w$ is not necessarily unbiased, as its unbiasedness depends on the specific allocation of sample sizes $n_i$, which is not guaranteed to be equal.
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
- Consider a finite population of size $N$. Let $T_1$ be the sample mean based on a sample of size $n$ under simple random sampling with replacement (SRSWR) scheme. Let $T_2$ be the sample mean based on a stratified random sample of size $n$ where the samples are drawn from each of 4 strata using SRSWR scheme under proportional allocation. Then which of the following are sufficient conditions for $\text{Var}(T_1) = \text{Var}(T_2)$ to hold?
- 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?
