Comprehensive Study Notes on Statistical Sampling Methods and Simple Random Sampling Techniques

Fundamentals of Sampling and Simple Random Sampling (SRS)

• Definition of Simple Random Sampling (SRS): Simple random sampling is a fair probability sampling method where every possible group of size $n$ from a population of size $N$ has the exact same chance of being picked, and every individual has an equal chance of being selected.

• Key Notation:

  • $N$ represents the total population size (uppercase letter).

  • $n$ represents the sample size (lowercase letter).

  • Strict Rule: The sample size $n$ must always be strictly less than the population size $N$ ($n < N$).

Advanced Probability and Combinatorics in SRS

• Combinatorial Enumeration: In SRS without replacement, the order of selection does not matter. To calculate the total number of distinct possible samples, use the combination formula:
  $\binom{N}{n}$

• Case Study (Sofia's Concert Tickets):

  • Scenario: Sofia has $4$ concert tickets. She keeps $1$ for herself and randomly picks $n = 3$ companions from $N = 6$ interested friends (Yolanda, Michael, Kevin, Marissa, Annie, and Katie).

  • Sample Space: There are exactly $\binom{6}{3} = 20$ unique, equally likely combinations of size $n = 3$:

  1. ${\text{Yolanda}, \text{Michael}, \text{Kevin}}$

  2. ${\text{Yolanda}, \text{Michael}, \text{Marissa}}$

  3. ${\text{Yolanda}, \text{Michael}, \text{Annie}}$

  4. ${\text{Yolanda}, \text{Michael}, \text{Katie}}$

  5. ${\text{Yolanda}, \text{Kevin}, \text{Marissa}}$

  6. ${\text{Yolanda}, \text{Kevin}, \text{Annie}}$

  7. ${\text{Yolanda}, \text{Kevin}, \text{Katie}}$

  8. ${\text{Yolanda}, \text{Marissa}, \text{Annie}}$

  9. ${\text{Yolanda}, \text{Marissa}, \text{Katie}}$

  10. ${\text{Yolanda}, \text{Annie}, \text{Katie}}$

  11. ${\text{Michael}, \text{Kevin}, \text{Marissa}}$

  12. ${\text{Michael}, \text{Kevin}, \text{Annie}}$

  13. ${\text{Michael}, \text{Kevin}, \text{Katie}}$

  14. ${\text{Michael}, \text{Marissa}, \text{Annie}}$

  15. ${\text{Michael}, \text{Marissa}, \text{Katie}}$

  16. ${\text{Michael}, \text{Annie}, \text{Katie}}$

  17. ${\text{Kevin}, \text{Marissa}, \text{Annie}}$

  18. ${\text{Kevin}, \text{Marissa}, \text{Katie}}$

  19. ${\text{Kevin}, \text{Annie}, \text{Katie}}$

  20. ${\text{Marissa}, \text{Annie}, \text{Katie}}$

  • Probability Calculation: Because each trio is equally likely, the probability of selecting any specific trio (e.g., Michael, Kevin, and Marissa) is:
    $P(\text{Specific Trio}) = \frac{1}{20} = 0.05 = 5\%$

Methods for Obtaining Simple Random Samples

• Population Frame: A frame is a complete list of all individuals in the population. Every individual must be assigned a unique numerical label from $1$ to $N$.

• Digit Consistency Rule: When using random digit tables, all labels must have the same number of digits as $N$.

  • Example: If $N = 30$, labels are $01$ to $30$.

  • Example: If $N = 500$, labels are $001$ to $500$.

• Sampling Protocols:

  • Sampling Without Replacement: Once selected, an individual is removed and cannot be picked again. The probability of selection for remaining individuals increases with each draw. This is the standard procedure in statistics.

  • Sampling With Replacement: Selected individuals are placed back into the pool and can be chosen multiple times. Probability remains constant at $\frac{1}{N}$.

• Implementation Tools:

  • Table of Random Digits:

  1. Pick an initial starting spot.

  2. Read numbers in blocks matching the label digit length.

  3. Ignore numbers exceeding $N$ or equal to $00$, and skip duplicates when sampling without replacement.

  4. Continue until $n$ distinct items are selected.

  • Sennesee and Associates Case Study:

  • Parameters: Population $N = 30$ clients labeled $01$ to $30$; sample size n = 5$.

  • Trace (Starting Row 13, Col 4-5):

    • Row 13, Col 4-5: 01$$\rightarrow$<strong>Selected</strong> (Client 01: ABC Electric)</p></li><li><p>Row 14, Col 4-5:$52$$\rightarrow$<strong>Ignore</strong> ($52 > 30$)</p></li><li><p>Row 15, Col 4-5:$07$$\rightarrow$<strong>Selected</strong> (Client 07: Dyno Jump)</p></li><li><p>Row 16, Col 4-5:$44$$\rightarrow$<strong>Ignore</strong> ($44 > 30$)</p></li><li><p>Row 17, Col 4-5:$46$$\rightarrow$<strong>Ignore</strong> ($46 > 30$)</p></li><li><p>Row 18, Col 4-5:$37$$\rightarrow$<strong>Ignore</strong> ($37 > 30$)</p></li><li><p>Transition to Col 6:$23$$\rightarrow Selected (Client 23: Travel Zone)

  • Graphing Calculators & Software:

  • Menu: MATH \rightarrow$<code>PROB</code></p></li><li><p><code>randInt(lower, upper, n)</code>: Generates$n$random integers (allows duplicates).</p></li><li><p><code>randIntNoRep(lower, upper, n)</code>: Generates$n$<strong>unique</strong> random integers (<strong>no duplicates</strong>).</p></li><li><p>Syntax Example: <code>randIntNoRep(1, 30, 5)</code> yields five unique selections like$5, 11, 4, 20, 29$.</p></li><li><p><strong>Physical Randomization:</strong> Writing labels on identical slips of paper, mixing thoroughly in a container, and drawing$n$slips.</p></li></ul></li></ul><h4>Stratified Sampling</h4><ul><li><p><strong>Definition:</strong> A <strong>stratified sample</strong> is obtained by dividing the population into separate, non-overlapping subgroups called <strong>strata</strong>, and then taking a <strong>simple random sample from EACH stratum</strong>.</p></li><li><p><strong>Stratum Homogeneity:</strong> Individuals within each stratum must be <strong>homogeneous (similar)</strong> regarding a specific characteristic (e.g., biological sex, age bracket, location).</p></li><li><p><strong>DePaul University Case Study:</strong></p><ul><li><p><strong>Objective:</strong> Survey campus safety opinions across population$N = 21\,909$.

  • Three Non-Overlapping Strata:

  1. Resident students: $N_1 = 6\,204$

  2. Nonresident / commuting students: $N_2 = 13\,304$

  3. Staff and faculty: $N_3 = 2\,401$

  • Proportional Sample Weighting ($n = 100$):

  • Resident students: $\frac{6\,204}{21\,909} \approx 28.31\% \rightarrow 28$ students

  • Nonresident students: $\frac{13\,304}{21\,909} \approx 60.72\% \rightarrow 61$ students

  • Staff / faculty: $\frac{2\,401}{21\,909} \approx 10.96\% \rightarrow 11$ staff members

  • Execution: Draw an SRS of $11$ staff, $28$ resident students, and $61$ nonresident students.

Systematic Sampling

• Definition: A systematic sample is obtained by selecting every $k^{\text{th}}$ individual from the population. The starting individual $p$ is randomly chosen between $1$ and k$.

• Procedure with Known Population Frame (N$):</strong></p><ol><li><p>Determine population size$N$and sample size$n$.

• Calculate interval size $k = \left\lfloor \frac{N}{n} \right\rfloor$. ALWAYS round DOWN to the nearest whole integer.

• Randomly choose starting integer $p$ between $1$ and $k$ inclusive.

• Sequence of selected labels: p, p+k, p+2k, p+3k, \dots, p+(n-1)k$.

• Procedure Without Frame (Unknown Population Size N$):</strong></p><ol><li><p>Pick an interval$k$(e.g., every$7^{\text{th}}$customer).</p></li><li><p>Choose a random start$p$between$1$and$k$.

• Sample every $k^{\text{th}}$ individual until sample size $n$ is reached.

• Detailed Case Studies:

  • Known Frame Example ($N = 100$, $n = 10$):

  • Interval $k = 4$, start p = 23$.

  • Sequence (10$individuals):$23, 27, 31, 35, 39, 43, 47, 51, 55, 59$.

  • Kroger Customer Satisfaction Example (Unknown $N$, $n = 40$):

  • Interval $k = 7$ (every $7^{\text{th}}$ customer), random start p = 5$.

  • Selected sequence: Customer 5, 12, 19, 26, \dots$</p></li><li><p>Formula for final ($n^{\text{th}}$) customer:<br>$p + (n-1)k$</p></li><li><p>Calculation:<br>$5 + (40-1) \times 7 = 5 + 39 \times 7 = 5 + 273 = 278$</p></li><li><p>Result: The <strong>$278^{\text{th}}$customer</strong> exiting the store is the final person surveyed.</p></li><li><p><strong>HR Benefits Survey Example ($N = 4\,408$,$n = 40$):</strong></p></li><li><p>Interval:$k = \frac{4\,408}{40} = 110.2 \rightarrow 110$(<strong>rounded down</strong>).</p></li><li><p>Random start:$p = 10$.

  • First three individuals: $10$, $10+110=120$, 10+2(110)=230$.

  • Final (40^{\text{th}}$) individual:<br>$10 + (40-1) \times 110 = 10 + 39 \times 110 = 10 + 4\,290 = 4\,300^{\text{th}}$employee.</p></li></ul></li></ul><h4>Cluster Sampling</h4><ul><li><p><strong>Definition:</strong> A <strong>cluster sample</strong> is obtained by selecting <strong>ALL individuals</strong> within a randomly selected group or collection of groups.</p></li><li><p><strong>Mechanism:</strong></p><ol><li><p>Divide the population into groups (<strong>clusters</strong>).</p></li><li><p>Take a simple random sample of the <strong>clusters</strong>.</p></li><li><p>Survey <strong>EVERY individual</strong> inside the selected clusters.</p></li></ol></li><li><p><strong>Primary Advantages:</strong> Saves significant time and money; <strong>does not require a list of every individual</strong> (only a frame of clusters).</p></li><li><p><strong>Boston Household Income Case Study:</strong></p><ul><li><p>Target: Household income across Boston city blocks.</p></li><li><p>Total city blocks (clusters):$10\,493$.

  • Procedure: Number blocks $1$ to $10\,493$. Draw an SRS of $20$ block numbers. Survey every household on those $20$ selected blocks.

Comparison of Sampling Techniques

• Stratified vs. Cluster Sampling Distinction:

  • Stratified Sampling: Divide population into strata $\rightarrow$ Take SRS from EACH stratum $\rightarrow$ Survey selected individuals. (NOT everyone in a stratum is surveyed).

  • Cluster Sampling: Divide population into clusters $\rightarrow$ Take SRS of the CLUSTERS $\rightarrow$ Survey ALL individuals in chosen clusters. (EVERYONE in chosen clusters is surveyed).

• Comparative Overview:

  • Simple Random Sampling: Pure chance selection from full list. Lowest bias, but high cost for widespread groups.

  • Stratified Sampling: Guarantees representation across key subgroups; best when opinions differ by subgroup.

  • Systematic Sampling: Interval selection ($k^{\text{th}}$). Best for continuous lines or assembly processes where a full frame is unknown.

  • Cluster Sampling: Whole group selection. Highly cost-efficient for geographically scattered populations.

Non-Probability Sampling: Convenience and Voluntary Response

• Convenience Sampling:

  • Mechanism: Individuals are selected based on easy accessibility without chance or randomization.

  • Example: Surveying the first $50$ students leaving a library to estimate total student study hours.

  • Flaws: Excludes online students and non-library visitors. Results are unrepresentative, heavily biased, and scientifically invalid.

• Voluntary Response Sampling:

  • Mechanism: Participants self-select whether to respond (e.g., call-in polls, text surveys, online forms).

  • Flaws: Strongly over-represents individuals with extreme or negative opinions. Highly biased and unreliable.

Multistage Sampling

• Definition: Multistage sampling combines two or more sampling techniques in successive steps. Used by professional research organizations for national data collection.

• Nielsen Media Research Case Study:

  • Objective: Track national TV viewing habits using electronic "people meter" devices.

  • Two-Stage Sampling Process:

  • Stage 1 (Stratified Sampling): Uses Census data to divide the U.S. into roughly $6\,000$ geographic strata (city blocks in urban areas, regional zones in rural areas).

  • Stage 2 (Simple Random Sampling): Lists households within selected strata and picks final households via simple random sampling.