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
- A simple random sample (without replacement) of size $n$ is drawn from a finite population of size $N (\ge 7)$. What is the probability that the $4^{\text{th}}$ population unit is included in the sample but the $6^{\text{th}}$ population unit is not included in the sample?
- For a data set $(x_1, y_1), (x_2, y_2), \dots, (x_n, y_n)$ the following two models were fitted using least square method.
Model 1: $y_i = \beta_0 + \beta_1 x_i \quad i = 1, 2, \dots n$
Model 2: $y_i = \beta_0^* + \beta_1^* x_i + \beta_2^* x_i^2 \quad i = 1, 2, \dots n$
Let $\hat{\beta}_0, \hat{\beta}_1$ be least square estimates of $\beta_0, \beta_1$ from model 1 and $\hat{\beta}_0^*, \hat{\beta}_1^*, \hat{\beta}_2^*$ be the least square estimates from model 2.
Let $A = \sum_1^n \left(y_i - (\hat{\beta}_0 + \hat{\beta}_1 x_i)\right)^2$,
$B = \sum_1^n \left(y_i - (\hat{\beta}_0^* + \hat{\beta}_1^* x_i + \hat{\beta}_2^* x_i^2)\right)^2$
Then - Suppose $\bar{Y}$ is the sample mean of the study variables corresponding to a sample of size n using simple random sampling with replacement scheme and $\bar{Y}_{st}$ is the sample mean of the study variables corresponding to a sample of size n using stratified random sampling with replacement scheme under proportional allocation. Which of the following is/are sufficient condition/conditions for $Var(\bar{Y}) = Var(\bar{Y}_{st})$?
- 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?
