Understanding Variance Estimation in SRSWOR

This question asks about finding the unbiased estimate of the variance of a sample proportion when using Simple Random Sampling Without Replacement (SRSWOR). Let's break down the concepts involved.

What is SRSWOR?

Simple Random Sampling Without Replacement (SRSWOR) is a method of selecting a sample of size 'n' from a population of size 'N' where each possible sample of size 'n' has an equal chance of being selected, and once a unit is selected, it is not returned to the population for further selection.

Key Terms:

• N: Total number of units in the population.
• n: Number of units selected in the sample.
• p: The proportion of sampled units having a specific attribute. This is the sample proportion.
• q: The proportion of sampled units *not* having the specific attribute, so \(q = 1 - p\).

Unbiased Estimate of Variance

When we estimate a population parameter (like the population proportion) using a sample statistic (like the sample proportion p), we also need to understand the variability of that statistic. The variance of the sample proportion, \(V(p)\), tells us how much the sample proportion is expected to vary from sample to sample. An unbiased estimate of this variance means that the average of the estimates obtained from all possible samples of the same size equals the true variance.

Deriving the Unbiased Variance Estimate

In SRSWOR, the variance of the sample mean \(\bar{y}\) is given by:

\(V(\bar{y}) = \left(\dfrac{N-n}{N-1}\right) \dfrac{\sigma^2}{n}\)

where \(\sigma^2\) is the population variance.

When dealing with proportions, we can think of the attribute as a binary variable (1 if the unit has the attribute, 0 otherwise). The population proportion P is the population mean, and the sample proportion p is the sample mean.

The population variance for a binary variable is \(P(1-P)\). An unbiased estimator of the population variance \(\sigma^2 = P(1-P)\) in SRSWOR is \(s^2 = \dfrac{n}{n-1} p(1-p)\).

So, the unbiased estimate of the variance of the sample mean (which is our sample proportion p) is obtained by replacing the population variance \(\sigma^2\) with its unbiased sample estimate \(s^2\) in the variance formula, and making a slight adjustment related to the finite population correction factor when using sample estimates:

The formula for the unbiased estimate of the variance of the sample proportion p in SRSWOR is:

\(v(p) = \left(\dfrac{N-n}{Nn}\right) s^2_y\)

where \(s^2_y\) is the sample variance for the binary variable, which is \(s^2_y = \dfrac{n}{n-1} p(1-p)\).

Substituting \(s^2_y\) into the formula for \(v(p)\):

\(v(p) = \left(\dfrac{N-n}{Nn}\right) \left(\dfrac{n}{n-1} p(1-p)\right)\)

Using \(q = 1-p\), this becomes:

\(v(p) = \left(\dfrac{N-n}{Nn}\right) \left(\dfrac{n}{n-1} pq\right)\)

We can simplify this expression:

\(v(p) = \dfrac{(N-n) \times n \times pq}{N \times n \times (n-1)}\)

\(v(p) = \dfrac{(N-n) pq}{N(n-1)}\)

This formula represents the unbiased estimate of the variance of the sample proportion p in SRSWOR.

Comparing with Options

Let's compare our derived formula with the given options:

• Option 1: \(\dfrac{pq}{(n-1)N}\) - This does not match.
• Option 2: pq / n - This is the variance formula for simple random sampling *with* replacement (SRSWR) or for an infinite population, using the sample proportion in the variance estimate, but it is not the unbiased estimate for SRSWOR.
• Option 3: N(N - n)pq / (n - 1) - This does not match.
• Option 4: \(\dfrac{(N - n)pq}{N(n-1)}\) - This exactly matches our derived formula.

Therefore, the unbiased estimate of \(V(p)\) in SRSWOR is \(\dfrac{(N - n)pq}{N(n-1)}\).

Revision Table: Key SRSWOR Concepts

Additional Information on Variance Estimation

Estimating the variance of a sample statistic is crucial in survey sampling because it allows us to calculate standard errors and construct confidence intervals. The standard error is the square root of the estimated variance. Confidence intervals provide a range of values within which the true population parameter is likely to lie.

The factor \(\dfrac{N-n}{N}\) is sometimes called the Finite Population Correction (FPC) factor, although strictly speaking, the FPC for the variance of the mean/proportion itself is \(\dfrac{N-n}{N-1}\). In the unbiased *estimate* of the variance, the factor appears as \(\dfrac{N-n}{N}\) multiplied by \(\dfrac{n}{n-1}\) times \(\dfrac{pq}{n}\), which simplifies to \(\dfrac{N-n}{N(n-1)} pq\). The term \(\dfrac{n}{n-1}\) acts as a correction factor for using the sample variance \(pq\) (based on \(n\) items) to estimate the population variance \(P(1-P)\), where the sample variance \(p(1-p)\) is typically defined with \(n\) in the denominator, and the unbiased estimate uses \(n-1\).

The use of \(n-1\) in the denominator for variance estimation is common in statistics (e.g., in sample variance \(s^2\)) to ensure the estimate is unbiased, especially when estimating a population variance from a sample. In the context of proportions, this correction appears indirectly in the unbiased variance estimate formula for SRSWOR.