Lecture 5: Sampling Techniques and Sampling Distributions

Foundations of Sampling and the Frame

• Population: The entire set of entities, people, or items of interest; descriptive measures computed on a population are called parameters ($\mu$, $\sigma$, $p$).

• Sample: A subset of the population from which data is gathered; descriptive measures computed on a sample are called statistics ($\bar{x}$, $s$, $\hat{p}$).

• Inferential Statistics: The practice of using sample data to make inferences and reach conclusions about the broader population.

• Census: Gathering data on the entire population.

  • Advantages of Sampling over a Census: Saves money, reduces time, broadens study scope for given resources, prevents destruction of products in destructive testing, and provides access when the full population is unreachable.

  • Reasons for a Census: Consumer safety, establishing benchmark data for future studies, and achieving complete representation without selection bias or random sampling error.

• Frame (Working Population): The operational list, directory, or map from which a sample is directly drawn.

  • Under-registered: The frame omits units that belong to the target population.

  • Over-registered: The frame contains units that do not belong to the target population.

  • Random samples statistically represent the frame, which can deviate from the true target population.

Random Sampling Techniques

• Random Sampling Requirement: Every unit has a well-defined probability of selection; standard inferential statistical methods assume and require random sampling where every unit has an equal selection probability.

• Simple Random Sampling: Every unit in the frame has an equal chance of being selected simultaneously; $n$ units are selected randomly without replacement (e.g., via random number generation).

• Systematic Sampling: Units are arranged in an ordered list, and every $k^{\text{th}}$ unit is selected.

  • Valid as a random sampling method provided there is no underlying periodicity in the data that aligns with interval $k$.

• Stratified Sampling: The population is partitioned into non-overlapping, mutually exclusive groups called strata, and independent random samples are taken from each stratum.

  • Used when strata differ substantially from one another and internal variation within each stratum is low.

  • Proportionate: Sample proportions for each stratum match their proportions in the overall population.

  • Disproportionate: Sample proportions differ from population proportions to ensure adequate representation of smaller strata.

• Cluster Sampling: The population is partitioned into clusters that serve as mini representations (microcosms) of the full population; a small number of clusters are randomly selected, and units within them are sampled completely or randomly.

  • Used primarily for cost reduction and geographic convenience.

• Stratified vs. Cluster Distinction:

  • Stratified: Samples are drawn from all strata; strata are internally homogeneous and externally heterogeneous.

  • Cluster: Samples are drawn from only some selected clusters; clusters are internally heterogeneous and externally homogeneous.

Nonrandom Sampling Techniques

• Nature of Nonrandom Sampling: Selection probabilities of population units are unknown or undefined; inferential statistical formulas cannot be mathematically applied.

• Convenience Sampling: Selection based on ease of access, proximity, or respondent willingness.

• Judgment Sampling: Selection based entirely on the researcher's subjective belief of representativeness.

• Quota Sampling: Stratification of the population where non-random, convenience-based methods are used to fill predefined quotas for each group.

• Snowball Sampling: Initial participants recruit or refer subsequent participants; applied when target populations are hard to identify or locate.

Statistical Errors and Biases

• Sampling Errors:

  • Random Sampling Error: Natural random variability among sample statistics due purely to chance selection of units.

  • Selection Bias: Systematic divergence between the sample and the population caused by non-random sample selection mechanisms.

• Non-Sampling Errors:

  • Measurement Error: Discrepancies between recorded observations and true actual values due to imprecise instruments or faulty collection tools.

  • Non-Response (Participation) Bias: Distortion occurring when response rates fall below 100% and respondents differ systematically from non-respondents.

  • Response Bias: Distortion occurring when respondents knowingly or unknowingly provide false or inaccurate answers.

  • Coverage Errors: Distortions caused by an under-registered or over-registered sampling frame.

Sampling Distributions and the Central Limit Theorem

• Sampling Distribution: The theoretical probability distribution of a sample statistic across all possible independent random samples of size $n$.

• Central Limit Theorem (CLT): For repeatedly drawn independent random samples of size $n$ from a population with mean $\mu$ and standard deviation $\sigma$, the distribution of sample means $\bar{x}$ is approximately normal if the sample size is sufficiently large ($n \ge 30$), regardless of the population distribution shape.

  • If the underlying population is normally distributed, the sampling distribution of $\bar{x}$ is normal for any sample size $n$.

• Mean of Sample Means: $\mu_{\bar{x}} = \mu$ (the sample mean $\bar{x}$ is an unbiased estimator of the population mean $\mu$).

• Standard Error of the Mean: The standard deviation of the sampling distribution: $\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}}$.

  • As sample size $n$ increases, the standard error decreases, producing a narrower, tighter distribution around $\mu$.

Inferences for Sample Means and Finite Populations

• Standardized $z$ Formula for Sample Means:

$z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}$

• Finite Population Correction (FPC): Used when sampling without replacement from a finite population of size $N$ where $N \le 20 \times n$:

$\sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} \sqrt{\frac{N - n}{N - 1}}$

$z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}} \sqrt{\frac{N - n}{N - 1}}}$

Sampling Distribution of Sample Proportions

• Sample Proportion ($\hat{p}$): The ratio of successes to total sample size, mathematically equivalent to the sample mean of a binary indicator variable coded as $1 = \text{true}$ and $0 = \text{false}$:

$\hat{p} = \frac{n(\text{true})}{n}$

• Parameters of the Sampling Distribution of $\hat{p}$:

  • Mean: $\mu_{\hat{p}} = p$

  • Standard Error: $\sigma_{\hat{p}} = \sqrt{\frac{p(1 - p)}{n}}$

• Normality Assumption for Proportions: The sampling distribution of $\hat{p}$ is approximately normal via the CLT if:

$n p > 5 \quad \text{and} \quad n(1 - p) > 5$

• Standardized $z$ Formula for Sample Proportions:

$z = \frac{\hat{p} - p}{\sqrt{\frac{p(1 - p)}{n}}}$