Sampling Methods and Sample Size Determination

Fundamental Concepts in Sampling

• Sampling Definition: The process of selecting a subset (sample) of a population to make inferences, estimate population parameters, and conduct statistical hypothesis tests.

• Reasons for Sampling: Reduced cost, greater speed/timeliness, greater efficiency and accuracy, greater scope, convenience, necessity (for destructive sampling), and ethical considerations (such as testing new drugs).

• Core Population & Frame Concepts:

  • Target Population: The population about which information is desired.

  • Sampled / Sampling Population: The population from which a sample is actually taken as determined by the sampling frame.

  • Sampling Frame: A complete list or mechanism providing observational access to sampling units in the population.

  • Under-coverage Error: Occurs when the sampling frame excludes units that are part of the target population.

  • Over-coverage Error: Occurs when the sampling frame includes units that are not part of the target population.

Classification of Sampling Techniques

• Sample Selection Criteria: Samples must be representative, provide precise and measurably reliable estimates, and minimize selection costs.

• Non-Probability Sampling:

  • Units are selected non-randomly; inclusion probabilities are unknown or zero for some units.

  • Restricted strictly to descriptive statements; cannot be used to make generalizations about the population.

  • Convenience / Accidental Sampling: Uses whichever units are readily available (e.g., selecting the first $100$ customers entering a store).

  • Judgment / Purposive Sampling: Selected according to the sampler's subjective judgment or intuition.

  • Quota Sampling: Arbitrarily selecting a specified number of units possessing given characteristics.

• Probability Sampling Methods:

  • Each population unit has a known, non-zero probability of inclusion, enabling valid statistical inference.

  • Simple Random Sampling (SRS): Every unit has an equal chance of inclusion. Conducted with or without replacement using random number tables, calculator RAN functions, chips-in-a-box, or statistical software.

  • Stratified Random Sampling: Population is divided into $L$ non-overlapping, homogeneous sub-populations (strata), and simple random samples are drawn independently from each stratum.

  • Equal Allocation: $n_1 = n_2 = \dots = n_h = \frac{n}{L}$

  • Proportional Allocation: $n_h = n \left( \frac{N_h}{N} \right)$

  • Systematic Random Sampling: Selecting every $k$th unit from an ordered frame starting from a random start $r$ ($1 \le r \le k$), where $k$ is the whole number ratio $\frac{N}{n}$ and $\frac{1}{k}$ is the sampling fraction.

  • Cluster Sampling: Population is partitioned into non-overlapping clusters containing heterogeneous elements. Sampling units are entire clusters, and all elements within selected clusters are enumerated. Sample size is not fixed.

Sample Size Determination

• Key Determinants:

  • Population Size ($N$): Total number of individuals in the demographic.

  • Margin of Error ($e$): Acceptable variation between the sample statistic and population parameter.

  • Confidence Level & Z-Score:

  • $90\%$ Confidence Level: $Z = 1.645$

  • $95\%$ Confidence Level: $Z = 1.96$

  • $99\%$ Confidence Level: $Z = 2.326$

  • Standard Deviation / Variance ($\sigma$ or $p$): Assumed as $p = 0.5$ for maximum variance when prior data is unavailable.

• Calculating Sample Size for Infinite or Very Large Populations:

  • Formula for estimating mean or proportion:     $n_0 = \frac{Z^2 \sigma^2}{e^2} \quad \text{and} \quad n_0 = \frac{Z^2 p(1 - p)}{e^2}$

  • Example ($95\%$ confidence level, $p = 0.5$, $e = 0.05$):     $n_0 = \frac{(1.96)^2 (0.5)(1 - 0.5)}{(0.05)^2} = \frac{(3.8416)(0.25)}{0.0025} = 384.16 \implies 385 \text{ respondents}$

• Finite Population Correction (FPC):

  • Sampling error formulas incorporating FPC:     $e = Z \frac{\sigma}{\sqrt{n}} \sqrt{\frac{N - n}{N - 1}}$     $e = Z \sqrt{\frac{p(1 - p)}{n}} \sqrt{\frac{N - n}{N - 1}}$

  • Adjusted sample size formula:     $n = \frac{n_0 N}{n_0 + (N - 1)}$

  • Saxon Home Improvement Company Examples ($N = 5000$):

  • Estimating Mean ($S = \$25$, $e = \$5$, $Z = 1.96$):       $n = \frac{(96.04)(5000)}{96.04 + (5000 - 1)} = 94.24 \implies 95$

  • Estimating Proportion ($p = 0.15$, $e = 0.07$, $Z = 1.96$):       $n = \frac{(99.96)(5000)}{99.96 + (5000 - 1)} = 98.02 \implies 99$

  • To satisfy both estimations simultaneously in a single sample, the larger sample size of $99$ is chosen.