Inference Reference
Sampling, Surveys, and Experiments
Reliable statistical conclusions start with a representative sample and a design that matches the claim being made.
Solved Examples
Jump directly to a worked example by study-design decision.
Fact Table
| Idea | Meaning | What It Supports |
|---|---|---|
| Population | Entire group of interest. | The group a conclusion aims to describe. |
| Random sample | Selection process gives population members a fair chance to be chosen. | Generalizing from sample to population. |
| Observational study | Researchers measure without assigning treatments. | Association, not causation. |
| Randomized experiment | Researchers assign treatments randomly. | Evidence for a causal effect within the study scope. |
Content Formulas
Sample Proportion
$$\hat p=\frac{\text{successes}}{\text{sample size}}$$
Margin of Error Idea
$$\text{estimate}\pm\text{margin of error}$$
Design Rule
$$\text{random sample}\Rightarrow\text{supports population inference}$$
Random assignment and random sampling answer different questions. Assignment supports causation; sampling supports generalization.
Classic Examples
Identify Population and Sample
A school randomly selects $50$ of its $800$ students to ask about lunch. Identify the population and sample.
Solution Steps
- Identify everyone the study wants to describe.
- Identify the students actually surveyed.
- State both groups.
$$\text{target group: all }800\text{ students}$$$$\text{surveyed group: }50\text{ selected students}$$$$\boxed{\text{population }=800,\quad\text{sample }=50}$$
Find Sampling Bias
A principal surveys only students in the chess club about school lunch. Explain the bias.
Solution Steps
- Identify who is excluded from the selection process.
- Explain why the group may differ from the full school.
- State the sampling problem.
$$\text{only chess-club students are sampled}$$$$\text{their lunch opinions may not represent all students}$$$$\boxed{\text{The sample is biased and not representative.}}$$
Classify a Study
Researchers randomly assign patients to receive a new medicine or a placebo. What type of study is this?
Solution Steps
- Identify that researchers assign treatments.
- Notice that the assignment is random.
- Classify the study and its causal strength.
$$\text{medicine or placebo is assigned}$$$$\text{assignment is random}$$$$\boxed{\text{randomized experiment; it can support causal inference}}$$
Evaluate a Claim
A poll of $200$ randomly selected voters finds $120$ support a proposal. Estimate the support proportion.
Solution Steps
- Use the number supporting the proposal as successes.
- Divide by the sample size.
- State the estimate as a sample result, not a certainty.
$$\hat p=\frac{120}{200}$$$$\hat p=0.60$$$$\boxed{\text{Estimated support is }60\%\text{ for the sampled population.}}$$