Choose the Inference Method from the Question and the Design
These lessons and problems are newly written for the revised course. Worked examples show complete reasoning; the application questions ask you to make and defend your own choices.
Begin with what one observation means
An inference procedure is a model for answering a population question using sample evidence. The first decision is what one observational unit contributes. A commuter supplies a yes/no answer to “uses the bus,” a category such as walk/bus/drive, or a quantitative time in minutes. Those responses may concern the same topic, but they target different parameters.
A binary response estimates a proportion p. A quantitative response estimates a mean μ. For two groups, specify the order of a difference such as p_A−p_B or μ_A−μ_B. For a multi-category distribution or an association between categorical variables, name the variables and the population distributions rather than forcing categories into a numerical mean. A numerical code assigned to “walk,” “bus” and “drive” does not make their average a meaningful commute-time parameter.
The second decision is the goal. “Estimate how many prefer the bus” calls for an interval, while “is support different from a baseline?” calls for a test. Write the actual parameter and population before selecting a formula. This prevents a sample percentage from being mistaken for a population hypothesis and prevents a mean time from being interpreted as a proportion.
Count independent samples, pairs and variables
Consider three complete transportation investigations. An SRS of 90 riders from 3,000 supplies one yes/no response per person. To estimate population bus support, choose a one-sample proportion z-interval. Two independent SRSs of 90 riders from town A and 110 from town B supply the same binary response. To test equal support proportions, choose a two-sample proportion z-test, with p_A and p_B defined for their respective rider populations.
Now suppose the same 18 randomly sampled riders record their journey time before and after a timetable change. The two columns are paired: a rider’s circumstances can affect both times. Analyze the ordered after-minus-before differences as one sample, target μ_d and use a one-sample paired t-test for evidence of mean change. Unknown population variability is estimated from the differences, not by pretending the columns are independent. By contrast, independent samples from two towns measuring journey time call for a two-sample t method for μ_A−μ_B.
A categorical table also needs its sampling story. Separate independent town samples with one commute category per person call for chi-square homogeneity. One SRS from one town with both dwelling type and commute category measured on each person calls for chi-square independence. The first compares distributions across populations; the second tests association between two categorical variables within one population. A table’s row labels alone do not identify its test.
Use a comparison table after understanding the design
| Observed response and design | Goal | Procedure and target |
|---|---|---|
| One binary response, one random sample | Estimate | One-sample z-interval for p |
| One binary response, one random sample | Test baseline | One-sample z-test for p |
| Binary response in two independent groups | Estimate difference | Two-sample z-interval for p_A−p_B |
| Binary response in two independent groups | Test equality | Two-sample z-test for p_A=p_B |
| One categorical variable, independent population samples | Compare distributions | Chi-square homogeneity |
| Two categorical variables, one population sample | Test association | Chi-square independence |
| Quantitative response, one sample, unknown σ | Estimate or test mean | One-sample t method for μ |
| Quantitative paired responses, unknown difference σ | Estimate or test mean change | One-sample t on differences for μ_d |
| Quantitative response, two independent groups, unknown σs | Estimate or test mean difference | Two-sample t method for μ_A−μ_B |
The table summarizes decisions; it does not replace a condition justification. After identifying a candidate method, explain the actual random selection/assignment, any without-replacement population fractions and the appropriate count or shape evidence. Proportion interval counts are observed; equality-null proportion test counts use the null value or pooled common estimate. Two-way chi-square requires every expected count strictly greater than 5 in this course. Mean t procedures need their actual sample-data route; paired small-sample shape concerns differences. A failed condition can make the proposed approximation unsupported even when a calculator accepts all entries.
A randomized experiment and a population survey also support different conclusions. Random assignment can support a causal treatment comparison within the design. Random sampling can support generalization to the population sampled. Volunteers assigned to treatments do not thereby become an SRS of all commuters.
Write a defensible method choice
For a worked mean comparison, independent SRSs of 45 devices from production line A and 50 from line B measure operating hours. Both population SDs are unknown and the question asks whether population mean lifetimes differ. Define μ_A and μ_B as mean operating lifetimes in hours in those production populations. Choose a two-sample t-test for independent means, with H₀:μ_A=μ_B and Hₐ:μ_A≠μ_B. A proportion z-test is incompatible because each device supplies a quantitative lifetime, not a binary success. Paired t is incompatible because different independent devices make up the groups. The next step would verify the specific population fractions and both sample-data conditions before calculating.
For a worked table comparison, three towns independently select random residents and record one commute choice for each. The question asks whether the town commute distributions are the same. Choose chi-square homogeneity, name the categorical response and all three town populations, then state the common-distribution null and not-all-same alternative. Independence would fit one population sample measuring two categorical variables, a different study. The expected-count check and full table are still needed before a calculation.
A useful written defense identifies the response type, study design, investigative goal and contextual parameter/variables. It also rejects a plausible incompatible procedure for a concrete reason. Complete the independent cases below using their own study details. Naming a method from a keyword alone is less informative than explaining why its model answers the particular question.
Independent Application
Work through each investigation before opening the separate solutions. State parameters, show calculations and connect your reasoning to the actual study.
Original generated practice for the Fall 2026 course.
Glossary and deeper reading · 13 refreshers
These editorial refreshers supplement the reading. Definitions and examples are added guidance; original source wording and recorded answers remain in their original context.
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Statistical inference
Statistical inference uses data and a study design to draw conclusions beyond the observed values. Random sampling supports population generalization, while well-conducted random assignment supports treatment-effect conclusions. For example, a random sample can estimate a school-wide preference without surveying every student.
Read more: Statistical inference in the captured Statistics reading →
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Population
A population is the full set of individuals or measurements a study aims to describe. It must be defined clearly before deciding whether a sample supports generalization. For example, all students enrolled in a particular school this year form a population.
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Mean
The mean is a measure of center often called an average. For a finite data set, the arithmetic mean is the sum of the values divided by their number; for a random variable, the mean is its probability-weighted expected value when that value exists. Means can be pulled toward extreme values and are not generally resistant. For example, the arithmetic mean of 2, 4, and 9 is 5.
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Categorical variable
A categorical variable records a group or label for each individual. Numeric labels still count as categories when arithmetic on them has no meaningful interpretation. For example, favorite school subject is categorical, even if each subject receives a numbered code.
Read more: Categorical variable in the captured Statistics reading →
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Population parameter
A population parameter is a numerical feature of the population or probability model being studied. It is fixed for that defined population, although its value may be unknown. For example, the true proportion of all enrolled students who walk to school is a parameter.
Read more: Population parameter in the captured Statistics reading →
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Sample
A sample is the subset of a population actually studied. How it is selected matters for the conclusions that can be drawn about the population. For example, 60 selected students may be a sample from a school’s full enrollment.
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Simple random sample
A simple random sample of a fixed size gives every possible subset of that size the same chance of selection. Giving individuals equal selection chances alone does not establish that the design is an SRS. For example, randomly choosing three identifiers without replacement from ten gives an SRS when every three-identifier set is equally likely.
Read more: Simple random sample in the captured Statistics reading →
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Sample proportion
A sample proportion is the number of sampled individuals with a specified characteristic divided by the sample size. It is commonly written p-hat and estimates the corresponding population proportion. For example, 18 walkers among 60 sampled students give a sample proportion of 0.30.
Read more: Sample proportion in the captured Statistics reading →
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Statistical independence
Statistical independence means that learning one outcome does not change the probability distribution of another. For two events, it implies that the probability of both occurring equals the product of their probabilities. For example, independent draws with replacement keep the second draw’s color probabilities unchanged by the first.
Read more: Statistical independence in the captured Statistics reading →
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Random sampling
Random sampling uses a specified chance process to select individuals from a population. It supports population inference when the sampling frame and data collection are appropriate, but does not guarantee perfect balance or eliminate every source of bias. For example, a random-number generator can select student identifiers from an enrollment list.
Read more: Random sampling in the captured Statistics reading →
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Expected count
An expected count is the count predicted for a category or cell under the null model. Goodness-of-fit expectations use the total count times the null proportion, while independence-table expectations use row total times column total divided by the grand total. For example, a null proportion of 0.2 in 50 trials gives an expected count of 10.
Read more: Expected count in the current two-way chi-square lesson →
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Experiment
An experiment deliberately imposes treatments and measures responses. Random assignment, comparison, and replication help distinguish treatment effects from other variation. For example, assigning plants to fertilizer treatments is an experiment.
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Survey
A survey collects information by asking individuals questions. Its quality depends on the sample design, question wording, and responses received. For example, a school can survey selected families about their transportation needs.