To determine the unbiased estimator for \(P(1 - P)\), where \(P\) represents the population proportion, let's analyze the concept of simple random sampling without replacement.
In simple random sampling without replacement, we draw a sample from a finite population without putting it back, which affects the sample statistics. Here's the step-by-step logic:
- The sample proportion \(p\) is given by the formula: \(p = \frac{x}{n}\), where \(x\) represents the number of units with the attribute in the sample.
- The expectation of the sample proportion is equal to the population proportion: \(\mathbb{E}[p] = P\).
- The variance of the sample proportion \(p\) in case of sampling without replacement is adjusted with a finite population correction factor, and is given by: \(\mathrm{Var}(p) = \frac{P(1-P)}{n} \cdot \frac{N-n}{N-1}\).
- To get an unbiased estimator for \(P(1-P)\), we adjust \(p(1-p)\) to match this structure. We need: \(\frac{n(N-1)}{N(n-1)} p(1-p)\).
Thus, the correct unbiased estimator for the variance of the sample proportion, which is \(P(1 - P)\) under simple random sampling without replacement, is:
\(\frac{n(N-1)}{N(n-1)} p(1 - p)\)
This accounts for the finite population correction factor which is necessary because the samples are drawn without replacement.
Therefore, the correct option is:
\(\frac{n(N-1)}{N(n-1)} p(1 - p)\)
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 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?
