Statistical Sampling: Mean & Total Estimation

A comprehensive analysis of estimators and sample size determination

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Key Insights

Unbiasedness of the Sample Mean: E(ȳ) = μ ensures that the sample mean is a reliable estimator for the population mean.
Finite Population Correction: Variance formulas incorporate factors like (N–n)/N to adjust for sampling without replacement.
Sample Size Determination: Derivations for both population mean and total require solving equations that balance desired error bounds with variance components.

A) Estimation of the Population Mean

Understanding the Estimator

1. Unbiasedness of the Sample Mean

When drawing a simple random sample of size n from a population of size N, we define the sample mean as:

ȳ = (1/n) ∑_i=1ⁿ y_i

The key property of the sample mean is its unbiasedness. In mathematical terms, the expected value of the sample mean is equal to the population mean:

E(ȳ) = μ

This result holds because every element y_i in the sample comes from a distribution with mean μ. By linearity of expectation,

E(ȳ) = E(1/n ∑ y_i) = (1/n) ∑ E(y_i) = (1/n) n μ = μ.

2. Variance of the Sample Mean When Population Variance Is Known

The variance of the sample mean, under the condition that the population variance σ² is known, is fundamental for understanding precision in estimation. When sampling is performed without replacement, the variance of ȳ is affected by a finite population correction (FPC) factor:

Var(ȳ) = [σ²/n] × [(N – n)/N]

The correction factor, (N – n)/N, reduces the variance estimate to account for the lack of independence when the sample is a large fraction of the entire population.

3. Variance of the Sample Mean When Population Variance Is Unknown

In many practical cases, σ² is not known and is estimated using the sample variance s². In this scenario, the variance of the sample mean becomes:

Var(ȳ) ≈ [s²/n] × [(N – n)/N]

Although s² is a random variable itself, replacing σ² with s² is common in applications, with the understanding that the precision of this replacement depends on the sample size.

4. Expected Value of the Sample Variance

When we examine the sample variance itself, its expected value has a slight bias relative to σ². For a simple random sample taken without replacement, it can be shown that:

E(s²) = [(N – 1)/N] σ²

This implies that the sample variance slightly underestimates the population variance. In many cases, a Bessel correction is applied (using n – 1 in the denominator rather than n) to yield an unbiased estimator for σ² when sampling from an infinite population.

5. Determining Sample Size for Estimating μ with a Specified Error Bound

Suppose we require the error of estimation of μ to be bounded by a given value. Denote by B the maximum acceptable bound on error. This error bound is connected to the variance of ȳ. Using the bound:

B = 2√[Var(ȳ)]

where we set:

Var(ȳ) = (s²/n) × [(N – n)/N] = B²/4

To derive the required sample size, let D = B²/4, then solving the equation:

(s²/n) × [(N – n)/N] = D

The algebraic manipulation to solve for n involves isolating n, resulting in:

n = [N σ²] / [σ² + N D]

This formula provides the sample size needed to ensure that the error in estimating μ does not exceed B.

B) Estimation of the Population Total

Estimator Construction and Variance Analysis

1. Estimator for the Population Total

The population total τ (tau) is defined as:

τ = N μ

A natural estimator for τ is obtained by “scaling up” the sample mean:

τ̂ = N ȳ

This estimator uses the proportionality between mean and total in the context of a finite population.

2. Variance of the Estimator τ̂

The variance of the total estimator can be derived by utilizing the variance of the sample mean. Given that τ̂ = N ȳ:

Var(τ̂) = N² Var(ȳ)

Substituting the expression for the variance of ȳ (with the finite population correction), we have:

Var(τ̂) = N² [σ²/n] × [(N – n)/N] = N σ² × [(N – n)/n]

Depending on whether σ² is known or estimated by s², the variance expression becomes:

Var(τ̂) ≈ N² [s²/n] × [(N – n)/N] = N s² × [(N – n)/n]

This quantity gives a direct measure of the variability in the estimator τ̂, which is crucial when assessing its precision.

3. Error Bound for Estimating τ

Similar to the estimation of μ, we can define a bound on the error of estimating the population total τ. If we denote the desired error bound by B, then this bound is related to the standard error of the estimator:

B = 2√[Var(τ̂)]

Substituting Var(τ̂):

B = 2√[N σ² (N – n)/n]

This provides a target for the variability that is acceptable in the estimator of the total.

4. Determining the Sample Size for Estimating τ

To derive the sample size n required for estimating τ with a prescribed error bound B, we start with:

N² (σ²/n) (1 - n/N) = B²

Rewriting this expression:

N² (σ²/n) [(N - n)/N] = N σ² (N - n)/n = B²

Let D represent B² as needed for the derivation (with the relation D = B² for the total context, or if a scaling such as B²/4 is used in parallel derivations for μ), then the equation becomes:

N σ² (N - n)/n = D

Solving for n, we rearrange the equation as follows:


    N σ² (N - n) = D n
    N² σ² - N σ² n = D n
    N² σ² = n (D + N σ²)
    n = (N² σ²) / (D + N σ²)

This formula provides the sample size required to bound the error in estimating the total τ at the predetermined level B. Adjustments might be necessary depending on whether the constant in front of B² is exactly 1 or follows a factor such as 1/4, as seen in different derivations.

Summary of Derived Expressions

Estimator/Expression	Formula	Description
E(ȳ)	μ	Unbiased estimator of the population mean
Var(ȳ)	$ \displaystyle \frac{\sigma^2}{n}\left(\frac{N-n}{N}\right) $	Variance of sample mean with finite population correction (FPC)
Var(ȳ) with unknown σ²	$ \displaystyle \frac{s^2}{n}\left(\frac{N-n}{N}\right) $	Using sample variance s² as an estimate
E(s²)	$ \displaystyle \frac{N-1}{N}\sigma^2 $	Expected value of the sample variance
Sample Size for μ	$ \displaystyle n = \frac{N\sigma^2}{\sigma^2 + N\left(\frac{B^2}{4}\right)} $	Required n so that error bound $ B = 2\sqrt{Var(ȳ)} $
Var(τ̂)	$ \displaystyle N^2\frac{\sigma^2}{n}\left(1-\frac{n}{N}\right) $	Variance of the population total estimator
Sample Size for τ	$ \displaystyle n = \frac{N^2\sigma^2}{D+N\sigma^2} $	Determined by setting $ N^2 (\sigma^2/n)(1-n/N) = B^2 $

Additional Mathematical Insights

Derivation Details Using Mathematical Expressions

Unbiasedness of ȳ

The unbiasedness of ȳ is succinctly shown using linearity of expectation:

$ \displaystyle E(ȳ) = E\Bigl(\frac{1}{n}\sum_{i=1}^{n} y_i\Bigr)= \frac{1}{n} \sum_{i=1}^{n} E(y_i)= \mu $

Variance Calculation with Finite Population Correction

When sampling without replacement, the covariance between different observations becomes non-zero. The rigorous derivation leads to:

$ \displaystyle Var(ȳ) = \frac{\sigma^2}{n} \left(1-\frac{n}{N}\right) $

This adjustment decreases the variance because the observations are not completely independent when the sample represents a significant portion of the population.