Stratified Random Sampling and Point Estimates Study Guide

Stratified Random Sampling and Point Estimates

Stratified random sampling is a probability sampling design in which the entire population is first divided into distinct, non-overlapping groups called strata. Each stratum is comprised of individuals who share similar, homogeneous attributes or characteristics relevant to the variable being measured. Once the strata are established, an independent Simple Random Sample (SRS) is selected from within each individual stratum, and the chosen individuals are combined to form the complete sample.

A high-quality sampling method strives to produce reliable point estimates. An effective sampling design achieves two main mathematical and statistical goals: minimizing systematic error to maintain low bias, and minimizing sample-to-sample fluctuations to achieve low variability. Stratifying by a relevant variable reduces sampling variability across different possible samples compared to a standard simple random sample.

Check Your Understanding: High School Digital Hall Pass Proposal

A principal at a large high school is proposing a new digital hall pass system that students would access using their smartphones. To gauge student opinion, the principal wishes to estimate the overall proportion of the student body in favor of this proposed change by conducting a survey on a sample size of $n = 100$ students.

Sampling Mechanism: Replacement vs. Without Replacement

When conducting this survey, the sampling must be executed without replacement. In sampling without replacement, once an individual student is selected for the sample, they cannot be chosen a second time.

Sampling without replacement is necessary because selecting the same student multiple times would yield duplicate response data without providing any additional unique information regarding the student population's overall perspective. Each response in the sample of size $n = 100$ must come from a distinct student to accurately reflect the diversity of opinions across the high school.

Procedure for Implementing a Stratified Random Sample

To obtain a stratified random sample of $n = 100$ students using grade level as the strata, the principal should execute the following step-by-step procedure:

First, divide the entire student population into four distinct strata based on grade level: 9th graders, 10th graders, 11th graders, and 12th graders.

Second, for the 9th-grade stratum, assign a unique numerical label from $1$ to $N_1$ to every student in that grade level, where $N_1$ represents the total population size of 9th graders.

Third, utilize a random number generator to select $25$ unique random integers between $1$ and $N_1$, ensuring that repeated numbers are ignored to maintain sampling without replacement.

Fourth, identify and select the $25$ 9th-grade students whose assigned numbers match the generated random integers.

Fifth, repeat this identical process for each of the remaining three grade-level strata (10th grade, 11th grade, and 12th grade), assigning unique numbers from $1$ to $N_i$ for each stratum, and selecting $25$ unique students from each grade level. Combining these four equal subsets yields the required stratified random sample of $100$ total students ($25 \times 4 = 100$).

Statistical Advantages of Stratified Random Sampling

A stratified random sample based on grade level produces a more precise estimate of the student body's opinion than a simple random sample because opinions regarding phone-based hall passes are likely to differ systematically between grade levels. For example, lowerclassmen such as 9th graders might overwhelmingly support a phone-based digital pass system, whereas upperclassmen such as 12th graders might largely oppose it.

In a simple random sample, purely by chance, the sample could end up over-representing one grade level and under-representing another (for instance, selecting an abnormally high percentage of 9th graders), leading to higher sample-to-sample variability in the estimated proportion. Stratifying guarantees that every grade level is proportionally and consistently represented in every possible sample, which directly reduces the overall variability of the point estimates and produces a more accurate, stable measure of the population parameter.