Stratified Sampling Methods

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Questions and Answers

What is the formula used to calculate the total sample size n for estimating the population with a specified cost C?

  • $ n = rac{W_i S_i^2}{ ext{sum}(c_i n_i)} $
  • $ n = rac{V + ext{sum}(W_i S_i^2 / N)}{ ext{sum}(W_i S_i / ext{sqrt}(c_i))} $
  • $ n = rac{(C-a)}{ ext{sum}(W_i S_i / ext{sqrt}(c_i))} $ (correct)
  • $ n = rac{C - a}{ ext{sum}(W_i S_i^2)} $

What relationship is established when minimizing both V' and C' according to the allocation results?

  • $ n_i ext{ is proportional to } ext{sqrt} rac{A_i c_i}{n_i} $ (correct)
  • $ n_i ext{ is proportional to } rac{c_i}{A_i n_i} $
  • $ n_i ext{ does not depend on C' } $
  • $ n_i ext{ is proportional to } A_i $

What condition must be met for equality in Cauchy-Schwartz's inequality when applied to the products V' and C'?

  • $ A_i $ must equal $ c_i $ for all $ i $
  • $ rac{c_i n_i}{A_i/n_i} $ must be constant across all strata (correct)
  • $ n_i $ must be equal for all strata
  • $ A_i c_i n_i $ remains constant for all $ i $

In the context of Neyman allocation, what assumption is made about A_i and c_i?

<p>A_i is W_i^2 and c_i is fixed. (A)</p> Signup and view all the answers

What does the formula $ n_i ext{ proportional to } ext{sqrt} rac{A_i c_i}{n_i} $ imply about the sample allocation?

<p>Sample sizes should increase with both A_i and c_i. (B)</p> Signup and view all the answers

What is the primary goal of sample size allocation in stratified sampling?

<p>Achieve maximum precision with minimum budget (D)</p> Signup and view all the answers

Which of the following is NOT a method of allocation in stratified sampling?

<p>Systematic allocation (D)</p> Signup and view all the answers

In proportional allocation, the sample size for each stratum is determined by which factor?

<p>Size of the stratum compared to total population size (D)</p> Signup and view all the answers

What does Neyman allocation aim to minimize?

<p>The variance for a fixed sample size (D)</p> Signup and view all the answers

Which formula represents the sampling fraction in proportional allocation?

<p>$ n_i = nN_i/N $ (D)</p> Signup and view all the answers

What does equal allocation imply in terms of sample distribution?

<p>Every stratum receives the same number of samples (D)</p> Signup and view all the answers

What is a characteristic of Neyman allocation concerning cost?

<p>Costs per unit sample are constant across strata (B)</p> Signup and view all the answers

Which allocation method is often chosen for administrative convenience?

<p>Equal allocation (A)</p> Signup and view all the answers

What is the main goal of the optimum allocation method in stratified sampling?

<p>To minimize the variance for a specific cost (A)</p> Signup and view all the answers

Which formula represents the cost function in stratified sampling?

<p>$ C = a + extstyle rac{ extstyle extstyle extstyle extstyle extstyle extstyle n_i}{ extstyle c_i} $ (B)</p> Signup and view all the answers

What is indicated when a stratum has larger variability according to the conclusions of relation (3.7.9)?

<p>A larger sample size is needed (B)</p> Signup and view all the answers

What does the variable $ eta $ represent in relation to the function $ ho = V(Ì„st) + eta C $?

<p>The constant related to the cost function (D)</p> Signup and view all the answers

In which situation does relation (3.7.9) lead to Neyman allocation?

<p>When costs are the same across strata (D)</p> Signup and view all the answers

What does the use of the calculus method of Lagrange multipliers allow researchers to do in stratified sampling?

<p>Determine optimum sample sizes under constraints (C)</p> Signup and view all the answers

What does the term 'regularized sampling' imply in the context of phased surveys?

<p>Sampling is adjusted based on initial data gathered (B)</p> Signup and view all the answers

If costs per unit vary among strata, how is the sample size determined according to relation (3.7.9)?

<p>It is larger for strata with lower costs (C)</p> Signup and view all the answers

Flashcards

Stratified Sampling Allocation

Distributing sample size across different groups (strata) based on factors like stratum size, variability, and cost.

Equal Allocation

Allocating the same sample size to each stratum.

Proportional Allocation

Allocating sample size proportionally to the size of each stratum.

Neyman Allocation

Allocating sample size based on the stratum size and variability; aims for minimum variance for a fixed budget.

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Stratum Size

The total number of units or people in a specific group (stratum).

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Stratum Variability

The degree of differences or spread within a particular group (stratum).

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Sampling Fraction (f)

Represents the ratio of sample size to total population size in each stratum in proportional allocation. It's uniform across all strata.

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Optimum Allocation

Allocating sample sizes that provide the maximum precision, using minimum resources.

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Optimum Allocation in Stratified Sampling

A method to determine sample sizes in each stratum (group) to minimize the variance of the sample mean or the cost of the survey, given a specific cost function.

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Cost Function (Stratified Sampling)

A function used to describe the total cost of a stratified sampling survey. It is often linear, where the total cost (C) is the sum of an overhead plus the cost per element in each stratum.

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Variance of Sample mean (Ï„st)

The measure of the spread of the sample mean taken from a stratified sampling survey.

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Sample Size Formula (náµ¢)

The formula for determining the optimal sample size for each stratum in an stratified sampling technique, where náµ¢ is proportional to the product of stratum size (Wáµ¢), stratum standard deviation (Sáµ¢), and inversely proportional to the square root of the cost per unit (cáµ¢).

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Stratum Variances

Variabilities within different sub-groups (strata) of a population, important for minimizing variance in stratified sampling.

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Lagrange Multipliers

Mathematical technique used in the optimization formula to find maximum or minimum of a function under constraints.

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Cauchy-Schwartz Inequality

A mathematical inequality that states the product of the sums of squares of two sequences is greater than or equal to the square of the sum of the products of the corresponding terms.

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V' and C'

Components of the total variance (V) and total cost (C) in stratified sampling that depend on the sample sizes within each stratum.

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Why minimize V’C’?

Minimizing the product of V' and C' ensures that the total variance is minimized for a given total cost, or vice versa.

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Sample Size Allocation Equation

The equation that determines the optimal sample size for each stratum based on the stratum variability, size, and cost.

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Study Notes

Stratified Sampling: Sample Size Allocation

  • Stratified sampling allocates samples to different strata based on factors like stratum size, variability, and cost of observation.
  • A good allocation maximizes precision with minimal resources. This means either maximizing precision for a given budget, or minimizing the budget for a given variance.
  • Four methods of sample size allocation in stratified sampling are:
    • Equal allocation: Dividing the total sample size equally among all strata.
    • Proportional allocation: Allocating samples to strata proportionally to their sizes within the population.
    • Neyman allocation: Allocating samples to strata based on stratum size multiplied by the stratum's variability divided by the overall cost.
    • Optimum allocation: Determining sample sizes based on minimizing the variance for a specified cost (or minimizing cost for a fixed variance).
    • Optimum allocation uses a formula to calculate the allocation (ni = (NiSi/√ci) * [Total sample Size/(ΣNiSi/√ci)]).
  • In optimum allocation, the overhead cost(a) and the average cost per unit in each stratum (ci) are constant. This method may require a pilot survey for estimating the stratum variance (Si).

Equal Samples From Each Stratum

  • A straightforward method for sample allocation from each stratum to make the variance smaller.

Proportional Allocation

  • Employs a simple method for proportionally allocating samples across strata to match the strata's portion in the overall population.

Neyman Allocation

  • Aims to minimize the variance by allocating more samples in strata with greater variability, ensuring the accuracy of the results.

Minimum Variance Allocation

  • Neyman Allocation is considered as a minimum variance method for allocating samples across strata to minimize variance within a fixed sample size. This method accounts for the variability within each stratum.

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Allocation of Sample Size in Different Strata PDF

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