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
- 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?
