Math 313: Survey Design and Sampling

Introduction to Estimating Proportions

Context: Estimating proportions in a population is key for decisions in marketing, public health, and policy making.
Example: Determining the proportion of voters favoring a particular policy, the proportion of households with internet access, or the percentage of people immunized in different regions.

Real-World Application

Marketing: A company may want to estimate the proportion of consumers who prefer a new product over existing ones. Stratified sampling can be used to ensure various demographics are represented.
Public Health: Health officials estimate the proportion of the population that has been vaccinated by stratifying the population by age, location, and other relevant factors.

Objective of Sample Allocation

Objective: Minimize the variance of the estimator while controlling costs.
Factors Influencing Allocation:
1. Population proportion in each stratum, $p_{i}$ .
2. Variability of the proportion, expressed as $p_{i} (1 - p_{i})$ .
3. Sampling cost per stratum, $c_{i}$ .

Step 1: Determine Weights for Sample Sizes

Weight Formula: $W_{i} = \frac{N_{i} \frac{\sqrt{p_{i} (1 - p_{i})}}{\sqrt{c_{i}}}}{\sum_{j = 1}^{L} N_{j} \frac{\sqrt{p_{j} (1 - p_{j})}}{\sqrt{c_{j}}}}$
Purpose: Adjust sample sizes to optimize precision and cost.

Step 2: Calculate Total Sample Size

Sample Size Formula: $n = \frac{\sum_{i = 1}^{L} N_{i}^{2} \frac{p_{i} (1 - p_{i})}{w_{i}}}{N^{2} D^{2} + \sum_{i = 1}^{L} N_{i} p_{i} (1 - p_{i})}$
Where:
- $D = \frac{B}{Z_{1 - \frac{α}{2}}}$ , defining the precision level for confidence intervals.

Step 3: Data Collection

Implementation: Collect data according to the sample sizes and weights determined in Steps 1 and 2.

Step 4: Estimation of Population Proportion

Estimator Formula: ${\hat{p}}_{s t} = \frac{1}{N} (N_{1} {\hat{p}}_{1} + N_{2} {\hat{p}}_{2} + \dots + N_{L} {\hat{p}}_{L})$
Variance of Estimator: $\hat{V} ({\hat{p}}_{s t}) = \frac{1}{N^{2}} \cdot \sum_{i = 1}^{L} N_{i}^{2} (1 - \frac{n_{i}}{N_{i}}) \cdot (\frac{{\hat{p}}_{i} (1 - {\hat{p}}_{i})}{n_{i} - 1})$

Degrees of Freedom for Variance Estimation

Degrees of Freedom Formula: $d f = \frac{{(\sum_{i = 1}^{L} a_{i} p_{i} (1 - p_{i}))}^{2}}{\sum_{i = 1}^{L} \frac{(a_{i} p_{i} (1 - p_{i}))^{2}}{n_{i} - 1}}$
Importance: Correct degrees of freedom are crucial for accurate inference and confidence interval construction.

Example 1

An advertising firm wants to estimate the proportion of households viewing TV show X in a county. The county is divided into three strata: town A, town B, and the rural area, containing $N_{1} = 155$ , $N_{2} = 62$ , and $N_{3} = 93$ households, respectively. From a study conducted three years ago, the previous estimates of proportions were ${\hat{p}}_{1}^{*} = 0.80$ , ${\hat{p}}_{2}^{*} = 0.25$ , and ${\hat{p}}_{3}^{*} = 0.50$ (these values are used for selecting sample sizes, not as the current proportions).

Costs: The costs of obtaining an observation are $c_{1} = c_{2} = 9$ , $c_{3} = 16$ .

The firm aims to estimate the population proportion $p$ with a 95% bound on the error of estimation equal to 0.1. The task is to find the total sample size $n$ and the strata sample sizes $n_{1}$ , $n_{2}$ , $n_{3}$ that will achieve the desired bound at minimum cost.
Based on the new survey performed with the determined sample sizes, the new sample proportions are ${\hat{p}}_{1} = 0.78$ , ${\hat{p}}_{2} = 0.35$ , ${\hat{p}}_{3} = 0.52$ . Estimate the proportion of households viewing TV show X with a 95% bound on the error of estimation.

Day 15