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 sample statistic describes:
- Worked Example: First identify exactly what the question is asking: A sample statistic describes:
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- A statistic is computed from sample data.
- A parameter describes the population.
Answer: the sample
Practice problems
1. Practice case A: A sample statistic describes:
Choices: the placebo · the null hypothesis · the sample · the entire population with certainty
Show solution
- Warm-up: First identify exactly what the question is asking: A sample statistic describes:
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- A statistic is computed from sample data.
- A parameter describes the population.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: the sample
2. Practice case B: Random sampling mainly helps a study:
Choices: make all variables categorical · generalize to a population · prove causation by itself · remove every outlier
Show solution
- Warm-up: First identify exactly what the question is asking: Random sampling mainly helps a study:
- 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 generalizing results.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: generalize to a population
3. Practice case C: Voluntary response samples are risky because:
Choices: people with strong opinions may be overrepresented · they are always too large · they have no variables · they prove causation
Show solution
- Warm-up: First identify exactly what the question is asking: Voluntary response samples are risky because:
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Voluntary samples depend on who chooses to respond.
- That can create bias.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: people with strong opinions may be overrepresented
4. Practice case D: Random assignment mainly helps an experiment:
Choices: choose a representative sample · increase every p-value · turn a table into a graph · compare treatments fairly
Show solution
- Warm-up: First identify exactly what the question is asking: Random assignment mainly helps an experiment:
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- Random assignment balances other factors across treatment groups.
- That supports fair treatment comparison.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: compare treatments fairly
5. Practice case E: A placebo is:
Choices: the population size · a residual from a model · a fake treatment used for comparison · the sample mean
Show solution
- Warm-up: First identify exactly what the question is asking: A placebo is:
- Compare each answer choice with the calculation or rule, and eliminate choices that do not satisfy the condition.
- A placebo looks like a treatment but lacks the active ingredient.
- It helps measure expectation effects.
- Verify the selected choice by checking that it satisfies the original prompt and that the other choices fail the same test.
Answer: a fake treatment used for comparison
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