Scope of Inference
A free Statistics and Data Analysis lesson from the “Collecting Data” unit, with a worked example and practice problems including step-by-step solutions.
Scope of inference asks what conclusion the design supports. Random sampling supports generalization to a population. Random assignment supports cause-and-effect conclusions. This lesson builds the habit of reading the context first, choosing the right statistical tool, calculating carefully, and then writing what the result means. By the end, students should be able to do the computation and explain why that computation answers the question.
What you'll learn
- Decide when a result can generalize
- Decide when a result can support causation
- Separate random sampling from random assignment
Worked example
Problem. A study randomly samples students from a school but does not assign any treatment. What conclusion is best supported?
- Worked Example: First identify exactly what the question is asking: A study randomly samples students from a school but does not assign any treatment. What conclusion is best supported?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Random sampling supports generalization to the sampled population.
- Without random assignment, causation is not strongly supported.
Answer: generalization to that school, not cause and effect
Practice problems
1. Practice case A: A survey randomly samples students from a school and records existing study time. What can it support?
Choices: both for every population · neither any conclusion nor any estimate · generalization, but not cause and effect · cause and effect only
Show solution
- Warm-up: First identify exactly what the question is asking: A survey randomly samples students from a school and records existing study time. What can it support?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Random sampling helps generalize.
- Observation alone does not establish causation.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: generalization, but not cause and effect
2. Practice case B: A teacher randomly assigns one class to two review methods. What can it best support?
Choices: the exact population parameter · cause and effect for similar participants, not broad generalization · broad population generalization only · neither comparison nor causation
Show solution
- Warm-up: First identify exactly what the question is asking: A teacher randomly assigns one class to two review methods. What can it best support?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Random assignment supports causation.
- A nonrandom sample limits generalization.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: cause and effect for similar participants, not broad generalization
3. Practice case C: A study randomly samples from a city and randomly assigns treatments. What can it support?
Choices: both generalization and cause and effect · only association · only the sample statistic · neither because randomness was used
Show solution
- Warm-up: First identify exactly what the question is asking: A study randomly samples from a city and randomly assigns treatments. What can it support?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Random sampling supports generalization.
- Random assignment supports cause and effect.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: both generalization and cause and effect
4. Practice case D: A voluntary online poll records existing habits. What is the main limit?
Choices: strong causation and broad generalization · a guaranteed exact parameter · automatic double blinding · limited generalization and no strong causal claim
Show solution
- Warm-up: First identify exactly what the question is asking: A voluntary online poll records existing habits. What is the main limit?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- No random sample limits generalization.
- No random assignment limits causation.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: limited generalization and no strong causal claim
5. Practice case E: Which part of design helps represent the target population?
Choices: placebo only · a small p-value · random sampling · random assignment
Show solution
- Warm-up: First identify exactly what the question is asking: Which part of design helps represent the target population?
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Random sampling helps the sample represent the population.
- That supports generalization.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: random sampling
Practice this interactively with instant feedback and an AI tutor.
Practice Scope of Inference Take the free placement check