AP Statistics Unit 1B Study Guide

1. The big picture: population, sample, and inference

Population: The entire group you want information about.

Sample: The individuals from whom data are actually collected.

Census: Data collected from every member of the population.

Parameter: A number describing a population.

Statistic: A number calculated from a sample; used to estimate a parameter.

Inference: Using sample results to draw a conclusion about a population.

Two kinds of randomness—know the difference

What is random?

What it supports

Random sampling- Generalizing results to the population

Random assignment- Making a cause-and-effect conclusion

Memory aid:Sample → generalize. Assignment → cause.

Random assignment does not automatically make a sample representative. A random sample does not establish causation by itself.

2. Investigative questions

Causation question-For [population], does [explanatory variable/treatment] cause an increase or decrease in [response variable], compared with [comparison group]?

Use causal language only when the study’s design supports it—typically through random assignment in an experiment.

Association question- For [population], is [explanatory variable] associated with a difference in [response variable], compared with [comparison group]?

Use association language for observational studies. Association alone does not show that one variable caused the other.

3. Observational studies vs. experiments

Observational study

Researchers observe or ask about variables; they do not assign treatments.

Can identify associations.

Cannot establish causation, because other variables may explain the relationship.

Example: Comparing SAT scores of students who chose to take a prep course with those who did not.

Experiment

Researchers impose treatments on experimental units and measure the response.

With random assignment, an experiment can support a cause-and-effect conclusion.

Random assignment helps balance the effects of lurking or confounding variables across treatment groups.

Identify the variables

Explanatory variable: The variable used to explain or predict changes.

Response variable: The outcome measured.

Treatment: A specific condition imposed in an experiment.

Experimental unit: The individual or object receiving a treatment. When units are people, they’re often called subjects.

Example: If students are randomly assigned to use a study app or not, then:

Explanatory variable: app use / treatment

Response variable: test score

Experimental units: students

4. Principles of a well-designed experiment

Comparison: Compare at least two treatments or a treatment with a control/placebo.

Random assignment: Assign experimental units to treatments by chance.

Replication: Use enough experimental units in each treatment group to distinguish real effects from chance variation.

Control: Keep other conditions the same or account for important variables.

Control group, placebo, and blinding

Control group: A comparison group that does not receive the active treatment, or receives a standard treatment.

Placebo: An inactive treatment designed to resemble the real one.

Placebo effect: A response caused by receiving a treatment or believing one was received, rather than by the active ingredient.

Blinding: Keeping participants and/or researchers unaware of treatment assignments to reduce bias.

Common experimental designs

Completely randomized design

Randomly assign all experimental units directly to treatment groups.

Best when units are reasonably similar, or no important grouping variable needs special control.

Randomized block design

First divide units into blocks based on a characteristic likely to affect the response.

Randomly assign treatments within each block.

Compare treatments while accounting for differences between blocks.

Example: Block by grade level, then randomly assign students in each grade to treatments.

Matched-pairs design

A special kind of block design.

Pair similar units and randomly assign one unit in each pair to each treatment; or have each unit receive both treatments in random order.

Compare results within pairs or within each individual.

Use blocking or matching when: You expect the grouping characteristic to affect the response. This can reduce variability and make treatment comparisons more precise.

5. Sampling methods

Simple random sample (SRS)

Every possible sample of the chosen size has an equal chance of being selected.

How to conduct: Label every individual, then use a random number generator to select the required number of unique labels.

Stratified random sample

Divide the population into strata based on a characteristic that may affect the response.

Take an SRS from every stratum.

Combine the selected individuals.

Why use it? Ensures representation from each group and can reduce variability when people within a stratum are similar.

Example: Sample students separately from each grade.

Cluster random sample

Divide the population into natural groups called clusters.

Randomly select some clusters.

Survey everyone in the selected clusters.

Why use it? Often cheaper or easier when groups are spread out.

Example: Randomly select several classrooms and survey every student in those classrooms.

Systematic random sample

Randomly select a starting point.

Select every \(k\)th individual after that.

Example: Choose a random starting room, then survey every 5th room.

Caution: A repeating pattern in the population that matches the sampling interval can bias the sample.

Quick distinction: stratified vs. cluster

Stratified: Sample some from every group.

Cluster: Sample all from some groups.

6. Sampling problems and bias

Bias is a systematic tendency to overestimate or underestimate the true population value. A large sample does not automatically eliminate bias.

Undercoverage

Some population members are missing or are less likely to be selected.

Example: Surveying only people listed in a phone directory misses people without listed numbers.

Nonresponse

Selected people cannot be reached or do not respond.

Example: Only 28 of 100 selected students return a survey.

Response bias

People give inaccurate answers, perhaps because of question wording, embarrassment, pressure, or interviewer influence.

Example: A leading question nudges respondents toward a particular answer.

Voluntary response bias

People choose themselves whether to participate; people with especially strong opinions may be more likely to respond.

Convenience sampling

Researchers sample individuals who are easiest to reach.

Example: Surveying people who happen to walk past one location.

Important: A sample can be large and still be biased. Increasing sample size reduces random sampling variability, but does not fix a biased sampling method.

7. How to write a valid conclusion

Before concluding, check what the design allows:

Was the sample randomly selected?If yes, results may be generalized to the population from which the sample was drawn.

Were treatments randomly assigned?If yes, a cause-and-effect conclusion may be justified for the experimental units studied.

Was there both random sampling and random assignment?Stronger basis for both generalizing to the population and making a causal conclusion.

Neither?Be cautious: results may describe the participants, but broad generalization and causal claims may not be justified.

Conclusion wording

Observational study: “There is an association between ___ and ___ among the individuals studied.”

Randomized experiment: “The treatment caused a difference in ___ for the experimental units studied.”

Random sample: “The results can be generalized to the population sampled.”

8. Common test traps

Saying “causes” after an observational study.

Saying a result represents the population just because the sample is large.

Confusing stratified (sample some from every group) with cluster (sample every member of some groups).

Calling a study an experiment when researchers did not assign treatments.

Forgetting to identify the population, not just the sample.

Saying random assignment allows generalization. It supports causal conclusions; random sampling supports generalization.

Giving a sampling method without explaining how to carry it out.

9. A reliable response template

For a study-design or sampling question, write:

Identify the population of interest.

Name the method or study type.

Describe the steps, including what is randomized.

State what the results can support: association, causation, generalization, or some combination.

Mention a limitation if relevant.

Example—stratified sample:“Divide all students into grade-level strata. Use a random number generator to select students from each grade, then combine the selected students. This ensures every grade is represented.”