Examining Sampling Probability Statements
The question asks to identify the statement that is always true for a sample of size $n \ (\ge 2)$ drawn without replacement from a finite population of size $N$ using an arbitrary sampling scheme.
Key Concepts:
- Inclusion Probability ($\pi_i$): The probability that the $i$-th unit is included in the sample.
- Joint Inclusion Probability ($\pi_{ij}$): The probability that both the $i$-th and $j$-th units are included in the sample.
Analysis of Options
Option 1: $\sum_{i=1}^N \pi_i = n$
This statement is a fundamental property of sampling with a fixed sample size. Let $I_i$ be the indicator variable for the inclusion of unit $i$ (1 if included, 0 otherwise). The inclusion probability is $\pi_i = P(I_i=1) = E[I_i]$. The total sample size $n$ is the sum of these indicator variables: $n = \sum_{i=1}^N I_i$. Using the linearity of expectation, the expected sample size is $E[n] = E[\sum_{i=1}^N I_i] = \sum_{i=1}^N E[I_i]$. Since $n$ is a fixed sample size, $E[n]=n$. Therefore, $n = \sum_{i=1}^N \pi_i$. This identity holds true for any sampling scheme that fixes the sample size at $n$.
Option 2: $\sum_{j=1, j \ne i}^N \pi_{ij} = n \pi_i$
This identity is not universally true for all arbitrary sampling schemes. While specific identities relating joint and individual inclusion probabilities exist, often in the context of variance estimation, this particular equation does not hold generally. It is known to be false for standard schemes like Simple Random Sampling Without Replacement (SRSWOR).
Option 3: $\pi_{ij} > 0$ for all $i < j$
This statement claims that every possible pair of distinct units $(i, j)$ has a non-zero probability of being selected together. This is not necessarily true. Depending on the chosen sampling scheme (e.g., certain stratified or systematic designs), it might be possible that $\pi_{ij} = 0$ for some pairs $(i, j)$. Therefore, this statement is not always true.
Option 4: $\pi_i \pi_j - \pi_{ij} > 0$
This inequality is equivalent to stating $\pi_i \pi_j > \pi_{ij}$. In sampling without replacement, selections are generally dependent, meaning $\pi_{ij} \le \pi_i \pi_j$. However, the strict inequality ($>$) is not always guaranteed. Equality ($\pi_i \pi_j = \pi_{ij}$) can occur under certain conditions or specific sampling designs, making the statement not *always* true.
Conclusion
Based on the analysis, the property $\sum_{i=1}^N \pi_i = n$ is the only statement guaranteed to be true for any arbitrary sampling scheme with a fixed sample size $n$ drawn without replacement from a finite population.
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 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?
