Sampling Methods: Simple Random, Stratified, Cluster, and Systematic Sampling
A lesson in probability sampling methods that moves from intuition and definitions to worked reasoning, error correction, and independent practice.
Lesson Goals: Sampling Methods
A probability sample requires a known chance mechanism applied to a frame; simple random, stratified, cluster, systematic, and multistage designs differ in how units are selected.
Population and sample
Population and sample in sampling methods: Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample.
Worked reasoning
For Population and sample in sampling methods, Design or critique a sampling plan for a campus dining survey that addresses population and sample.
When the idea is valid
For Population and sample in sampling methods, The random mechanism must operate on units in a frame that adequately covers the target population.
Misconception to remove
For Population and sample in sampling methods, reject this error: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Census
Census in sampling methods: Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample.
Worked reasoning
For Census in sampling methods, Design or critique a sampling plan for a seedling-growth comparison that addresses census.
When the idea is valid
For Census in sampling methods, The random mechanism must operate on units in a frame that adequately covers the target population.
Misconception to remove
For Census in sampling methods, reject this error: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Simple random sample
Simple random sample in sampling methods: Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample.
Worked reasoning
For Simple random sample in sampling methods, Design or critique a sampling plan for a school library checkout study that addresses simple random sample.
When the idea is valid
For Simple random sample in sampling methods, The random mechanism must operate on units in a frame that adequately covers the target population.
Misconception to remove
For Simple random sample in sampling methods, reject this error: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Stratified
Stratified in sampling methods: Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample.
Worked reasoning
For Stratified in sampling methods, Design or critique a sampling plan for a city bus arrival investigation that addresses stratified.
When the idea is valid
For Stratified in sampling methods, The random mechanism must operate on units in a frame that adequately covers the target population.
Misconception to remove
For Stratified in sampling methods, reject this error: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Cluster
Cluster in sampling methods: Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample.
Worked reasoning
For Cluster in sampling methods, Design or critique a sampling plan for a quality-control inspection that addresses cluster.
When the idea is valid
For Cluster in sampling methods, The random mechanism must operate on units in a frame that adequately covers the target population.
Misconception to remove
For Cluster in sampling methods, reject this error: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Systematic
Systematic in sampling methods: Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample.
Worked reasoning
For Systematic in sampling methods, Design or critique a sampling plan for a quality-control inspection that addresses systematic.
When the idea is valid
For Systematic in sampling methods, The random mechanism must operate on units in a frame that adequately covers the target population.
Misconception to remove
For Systematic in sampling methods, reject this error: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Multistage
Multistage in sampling methods: Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample.
Worked reasoning
For Multistage in sampling methods, Design or critique a sampling plan for a commuter route study that addresses multistage.
When the idea is valid
For Multistage in sampling methods, The random mechanism must operate on units in a frame that adequately covers the target population.
Misconception to remove
For Multistage in sampling methods, reject this error: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Convenience and voluntary response
Convenience and voluntary response in sampling methods: Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample.
Worked reasoning
For Convenience and voluntary response in sampling methods, Design or critique a sampling plan for a greenhouse germination experiment that addresses convenience and voluntary response.
When the idea is valid
For Convenience and voluntary response in sampling methods, The random mechanism must operate on units in a frame that adequately covers the target population.
Misconception to remove
For Convenience and voluntary response in sampling methods, reject this error: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Comparison table
Comparison table in sampling methods: Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample.
Worked reasoning
For Comparison table in sampling methods, Design or critique a sampling plan for a campus dining survey that addresses comparison table.
When the idea is valid
For Comparison table in sampling methods, The random mechanism must operate on units in a frame that adequately covers the target population.
Misconception to remove
For Comparison table in sampling methods, reject this error: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Guided, Independent and Challenge Practice
Every question in Sampling Methods: Simple Random, Stratified, Cluster, and Systematic Sampling is newly written from the revised framework and the logic visible in public College Board materials. Constructed numerical settings are identified as instructional scenarios and are never represented as measurements from a real population. No released or secure question wording is reproduced.
Easy Practice
Easy 1: Multistage
Question P36-Easy-1. Design or critique a sampling plan for a classroom memory study that addresses multistage.
Worked solution and validity check
Worked solution P36-Easy-1. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Easy 2: Convenience and voluntary response
Question P36-Easy-2. Design or critique a sampling plan for a public-parks visitor survey that addresses convenience and voluntary response.
Worked solution and validity check
Worked solution P36-Easy-2. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Easy 3: Comparison table
Question P36-Easy-3. Design or critique a sampling plan for a battery-life laboratory trial that addresses comparison table.
Worked solution and validity check
Worked solution P36-Easy-3. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Easy 4: Population and sample
Question P36-Easy-4. Design or critique a sampling plan for a classroom memory study that addresses population and sample.
Worked solution and validity check
Worked solution P36-Easy-4. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Easy 5: Census
Question P36-Easy-5. Design or critique a sampling plan for a commuter route study that addresses census.
Worked solution and validity check
Worked solution P36-Easy-5. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Easy 6: Simple random sample
Question P36-Easy-6. Design or critique a sampling plan for a manufacturing fill-volume check that addresses simple random sample.
Worked solution and validity check
Worked solution P36-Easy-6. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Easy 7: Stratified
Question P36-Easy-7. Design or critique a sampling plan for a reading-speed investigation that addresses stratified.
Worked solution and validity check
Worked solution P36-Easy-7. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Easy 8: Cluster
Question P36-Easy-8. Design or critique a sampling plan for a classroom memory study that addresses cluster.
Worked solution and validity check
Worked solution P36-Easy-8. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Easy 9: Systematic
Question P36-Easy-9. Design or critique a sampling plan for a city bus arrival investigation that addresses systematic.
Worked solution and validity check
Worked solution P36-Easy-9. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Easy 10: Multistage
Question P36-Easy-10. Design or critique a sampling plan for a water-filtration experiment that addresses multistage.
Worked solution and validity check
Worked solution P36-Easy-10. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Easy 11: Convenience and voluntary response
Question P36-Easy-11. Design or critique a sampling plan for a recycling-behavior survey that addresses convenience and voluntary response.
Worked solution and validity check
Worked solution P36-Easy-11. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Easy 12: Comparison table
Question P36-Easy-12. Design or critique a sampling plan for a classroom memory study that addresses comparison table.
Worked solution and validity check
Worked solution P36-Easy-12. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Easy 13: Population and sample
Question P36-Easy-13. Design or critique a sampling plan for a manufacturing fill-volume check that addresses population and sample.
Worked solution and validity check
Worked solution P36-Easy-13. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Easy 14: Census
Question P36-Easy-14. Design or critique a sampling plan for a city bus arrival investigation that addresses census.
Worked solution and validity check
Worked solution P36-Easy-14. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Easy 15: Simple random sample
Question P36-Easy-15. Design or critique a sampling plan for a water-filtration experiment that addresses simple random sample.
Worked solution and validity check
Worked solution P36-Easy-15. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Easy 16: Stratified
Question P36-Easy-16. Design or critique a sampling plan for a school library checkout study that addresses stratified.
Worked solution and validity check
Worked solution P36-Easy-16. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Tough Practice
Tough 1: Stratified
Question P36-Tough-1. Design or critique a sampling plan for a reading-speed investigation that addresses stratified.
Worked solution and validity check
Worked solution P36-Tough-1. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Tough 2: Cluster
Question P36-Tough-2. Design or critique a sampling plan for a battery-life laboratory trial that addresses cluster.
Worked solution and validity check
Worked solution P36-Tough-2. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Tough 3: Systematic
Question P36-Tough-3. Design or critique a sampling plan for a water-filtration experiment that addresses systematic.
Worked solution and validity check
Worked solution P36-Tough-3. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Tough 4: Multistage
Question P36-Tough-4. Design or critique a sampling plan for a water-filtration experiment that addresses multistage.
Worked solution and validity check
Worked solution P36-Tough-4. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Tough 5: Convenience and voluntary response
Question P36-Tough-5. Design or critique a sampling plan for a package-delivery sample that addresses convenience and voluntary response.
Worked solution and validity check
Worked solution P36-Tough-5. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Tough 6: Comparison table
Question P36-Tough-6. Design or critique a sampling plan for a package-delivery sample that addresses comparison table.
Worked solution and validity check
Worked solution P36-Tough-6. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Tough 7: Population and sample
Question P36-Tough-7. Design or critique a sampling plan for a public-parks visitor survey that addresses population and sample.
Worked solution and validity check
Worked solution P36-Tough-7. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Tough 8: Census
Question P36-Tough-8. Design or critique a sampling plan for a public-parks visitor survey that addresses census.
Worked solution and validity check
Worked solution P36-Tough-8. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Tough 9: Simple random sample
Question P36-Tough-9. Design or critique a sampling plan for a commuter route study that addresses simple random sample.
Worked solution and validity check
Worked solution P36-Tough-9. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Tough 10: Stratified
Question P36-Tough-10. Design or critique a sampling plan for a water-filtration experiment that addresses stratified.
Worked solution and validity check
Worked solution P36-Tough-10. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Tough 11: Cluster
Question P36-Tough-11. Design or critique a sampling plan for a seedling-growth comparison that addresses cluster.
Worked solution and validity check
Worked solution P36-Tough-11. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Tough 12: Systematic
Question P36-Tough-12. Design or critique a sampling plan for a battery-life laboratory trial that addresses systematic.
Worked solution and validity check
Worked solution P36-Tough-12. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Tough 13: Multistage
Question P36-Tough-13. Design or critique a sampling plan for a battery-life laboratory trial that addresses multistage.
Worked solution and validity check
Worked solution P36-Tough-13. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Tough 14: Convenience and voluntary response
Question P36-Tough-14. Design or critique a sampling plan for a tutoring-program evaluation that addresses convenience and voluntary response.
Worked solution and validity check
Worked solution P36-Tough-14. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Tough 15: Comparison table
Question P36-Tough-15. Design or critique a sampling plan for a classroom memory study that addresses comparison table.
Worked solution and validity check
Worked solution P36-Tough-15. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Tough 16: Population and sample
Question P36-Tough-16. Design or critique a sampling plan for a tutoring-program evaluation that addresses population and sample.
Worked solution and validity check
Worked solution P36-Tough-16. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Toughest Practice
Toughest 1: Systematic
Question P36-Toughest-1. Design or critique a sampling plan for an online-course completion sample that addresses systematic.
Worked solution and validity check
Worked solution P36-Toughest-1. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Toughest 2: Multistage
Question P36-Toughest-2. Design or critique a sampling plan for a seedling-growth comparison that addresses multistage.
Worked solution and validity check
Worked solution P36-Toughest-2. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Toughest 3: Convenience and voluntary response
Question P36-Toughest-3. Design or critique a sampling plan for a quality-control inspection that addresses convenience and voluntary response.
Worked solution and validity check
Worked solution P36-Toughest-3. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Toughest 4: Comparison table
Question P36-Toughest-4. Design or critique a sampling plan for a tutoring-program evaluation that addresses comparison table.
Worked solution and validity check
Worked solution P36-Toughest-4. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Toughest 5: Population and sample
Question P36-Toughest-5. Design or critique a sampling plan for a public-parks visitor survey that addresses population and sample.
Worked solution and validity check
Worked solution P36-Toughest-5. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Toughest 6: Census
Question P36-Toughest-6. Design or critique a sampling plan for a package-delivery sample that addresses census.
Worked solution and validity check
Worked solution P36-Toughest-6. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Toughest 7: Simple random sample
Question P36-Toughest-7. Design or critique a sampling plan for a manufacturing fill-volume check that addresses simple random sample.
Worked solution and validity check
Worked solution P36-Toughest-7. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Toughest 8: Stratified
Question P36-Toughest-8. Design or critique a sampling plan for a school library checkout study that addresses stratified.
Worked solution and validity check
Worked solution P36-Toughest-8. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Toughest 9: Cluster
Question P36-Toughest-9. Design or critique a sampling plan for a tutoring-program evaluation that addresses cluster.
Worked solution and validity check
Worked solution P36-Toughest-9. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Toughest 10: Systematic
Question P36-Toughest-10. Design or critique a sampling plan for a seedling-growth comparison that addresses systematic.
Worked solution and validity check
Worked solution P36-Toughest-10. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Toughest 11: Multistage
Question P36-Toughest-11. Design or critique a sampling plan for a package-delivery sample that addresses multistage.
Worked solution and validity check
Worked solution P36-Toughest-11. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Toughest 12: Convenience and voluntary response
Question P36-Toughest-12. Design or critique a sampling plan for a city bus arrival investigation that addresses convenience and voluntary response.
Worked solution and validity check
Worked solution P36-Toughest-12. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Toughest 13: Comparison table
Question P36-Toughest-13. Design or critique a sampling plan for a commuter route study that addresses comparison table.
Worked solution and validity check
Worked solution P36-Toughest-13. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Toughest 14: Population and sample
Question P36-Toughest-14. Design or critique a sampling plan for an online-course completion sample that addresses population and sample.
Worked solution and validity check
Worked solution P36-Toughest-14. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Toughest 15: Census
Question P36-Toughest-15. Design or critique a sampling plan for a manufacturing fill-volume check that addresses census.
Worked solution and validity check
Worked solution P36-Toughest-15. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
Toughest 16: Simple random sample
Question P36-Toughest-16. Design or critique a sampling plan for a classroom memory study that addresses simple random sample.
Worked solution and validity check
Worked solution P36-Toughest-16. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. Interpretation: Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. Validity: The random mechanism must operate on units in a frame that adequately covers the target population. Error to reject: Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame.
AP Response and Publication Checklist
| Audit point | Required evidence for sampling methods |
|---|---|
| Scope | Own selection mechanics; P37 owns bias diagnosis. |
| Method or source | A probability sample requires a known chance mechanism applied to a frame; simple random, stratified, cluster, systematic, and multistage designs differ in how units are selected. |
| Calculation | For a systematic sample of 80 from a frame of 2,400, use interval , choose a random start from 1 through 30, then select every 30th unit. |
| Interpretation | Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. |
| Validity | The random mechanism must operate on units in a frame that adequately covers the target population. |
| Correction | Selecting the first 80 convenient units is not systematic random sampling because there is no random start and fixed interval over the frame. |
Frequently Asked Questions
How does population and sample work in sampling methods?
Answer for sampling methods and Population and sample. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. The required validity evidence is: The random mechanism must operate on units in a frame that adequately covers the target population.
How does census work in sampling methods?
Answer for sampling methods and Census. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. The required validity evidence is: The random mechanism must operate on units in a frame that adequately covers the target population.
How does simple random sample work in sampling methods?
Answer for sampling methods and Simple random sample. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. The required validity evidence is: The random mechanism must operate on units in a frame that adequately covers the target population.
How does stratified work in sampling methods?
Answer for sampling methods and Stratified. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. The required validity evidence is: The random mechanism must operate on units in a frame that adequately covers the target population.
How does cluster work in sampling methods?
Answer for sampling methods and Cluster. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. The required validity evidence is: The random mechanism must operate on units in a frame that adequately covers the target population.
How does systematic work in sampling methods?
Answer for sampling methods and Systematic. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. The required validity evidence is: The random mechanism must operate on units in a frame that adequately covers the target population.
How does sample sampling method connect to Sampling Methods?
sample sampling method within sampling methods. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. For Population and sample, the controlling scope is: Own selection mechanics; P37 owns bias diagnosis.
How does types of sampling methods connect to Sampling Methods?
types of sampling methods within sampling methods. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. For Census, the controlling scope is: Own selection mechanics; P37 owns bias diagnosis.
How does kinds of sampling method connect to Sampling Methods?
kinds of sampling method within sampling methods. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. For Simple random sample, the controlling scope is: Own selection mechanics; P37 owns bias diagnosis.
How does methods of sampling statistics connect to Sampling Methods?
methods of sampling statistics within sampling methods. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. For Stratified, the controlling scope is: Own selection mechanics; P37 owns bias diagnosis.
How does sampling and sampling methods connect to Sampling Methods?
sampling and sampling methods within sampling methods. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. For Cluster, the controlling scope is: Own selection mechanics; P37 owns bias diagnosis.
How does sampling method connect to Sampling Methods?
sampling method within sampling methods. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. For Systematic, the controlling scope is: Own selection mechanics; P37 owns bias diagnosis.
How does sampling method techniques connect to Sampling Methods?
sampling method techniques within sampling methods. Define the population and frame, use a chance mechanism, identify the selected units, and anticipate undercoverage, nonresponse, and measurement error. Random selection supports probability-based inference to the covered population; it does not guarantee a representative realized sample. For Multistage, the controlling scope is: Own selection mechanics; P37 owns bias diagnosis.
Sources
Administrative and curricular statements in Sampling Methods: Simple Random, Stratified, Cluster, and Systematic Sampling were checked on July 18, 2026. The linked College Board pages control any later policy change; all instructional datasets in original questions are explicitly constructed rather than attributed to a real study.
Sampling Methods Conclusion
A probability sample requires a known chance mechanism applied to a frame; simple random, stratified, cluster, systematic, and multistage designs differ in how units are selected. Mastery of sampling methods therefore requires the exact evidence, mathematics, interpretation, and scope developed in this guide, while preserving this boundary: Own selection mechanics; P37 owns bias diagnosis.
