Choose From the Question, Not Just the Story
Inference questions may share a setting but require different procedures. A question about the percentage of residents who support a proposal concerns a proportion; a question about average commute time concerns a mean. Even when two groups are compared, the right method depends on whether the measurements are categorical or quantitative and whether observations are independent or paired.
Earlier in this course, you learned the procedures themselves. The new skill here is selecting among them reliably. Start with the response variable and the parameter named in the question. Then identify the number and structure of the samples or groups. Finally, check that the chosen procedure’s conditions fit the data-collection design and data.
A Sorting Routine for Inference Questions
A “group” in a story does not automatically mean a two-sample procedure. For example, the same people measured before and after an intervention form matched pairs, not two independent samples. Likewise, a table of counts for two categorical variables calls for a chi-square procedure rather than a comparison of means.
A categorical response places each individual in a category, such as yes or no. A quantitative response is a numerical measurement, such as time, distance, or score.
Look for a population proportion \(p\), a population mean \(\mu\), a difference in proportions or means, or an association between categorical variables.
Count the samples or groups, and decide whether observations are independent or linked in pairs. For categorical data, ask whether the question concerns one proportion, a difference in two proportions, or a relationship in a two-way table.
Consider how the data were collected, whether observations can reasonably be treated as independent, and whether the applicable Large Counts or distribution condition is met.
Match the Data Structure to the Procedure
For one sample with a binary categorical response, the parameter is usually a population proportion \(p\). A one-proportion \(z\) test addresses a claim about \(p\); a one-proportion \(z\) interval estimates \(p\). For two independent samples or independently assigned groups with binary outcomes, use a two-proportion \(z\) procedure to compare population proportions. The parameter is then \(p_1-p_2\).
For a quantitative response, one sample leads to inference about a population mean \(\mu\), using a one-sample \(t\) procedure. Two independent groups lead to a two-sample \(t\) procedure for comparing population means. If the same individuals are measured twice, or if observations are deliberately matched, calculate a difference for each pair and use a paired \(t\) procedure. The parameter is the population mean difference, \(\mu_d\).
Chi-square procedures use counts in categories, not quantitative measurements or individual percentages treated as though they were counts. A chi-square test of independence assesses whether two categorical variables are associated in one population. A chi-square test of homogeneity compares the distribution of one categorical variable across two or more populations or groups. The data layout may look similar; the question and sampling design determine which name fits.
| Question and data structure | Procedure family |
|---|---|
| One sample; one binary categorical outcome; one population proportion | One-proportion \(z\) |
| Two independent samples or groups; binary categorical outcome; compare proportions | Two-proportion \(z\) |
| One sample; quantitative measurement; one population mean | One-sample \(t\) |
| Two independent samples or groups; quantitative measurement; compare means | Two-sample \(t\) |
| Matched observations; quantitative measurement; analyze within-pair differences | Paired \(t\) |
| Counts for categorical variables in a table | Chi-square test of independence or homogeneity |
Selecting the procedure is not the same as deciding whether all its conditions are met. For example, a one-proportion \(z\) test is the right family for a single binary response and a claim about \(p\), but it still requires appropriate data collection and the Large Counts condition using the null proportion. An interval and a test may have different Large Counts checks.
Conditions Help Confirm the Choice
A procedure’s conditions serve two purposes: they help establish whether the method is appropriate, and they identify what evidence a written solution should report. State the relevant conditions explicitly rather than assuming that a large sample or a familiar setting makes them automatic.
For a paired \(t\) procedure, the relevant distribution condition applies to the differences, not separately to the two lists of measurements. For a two-sample \(t\) procedure, the two samples or groups must be independent of each other; a large total sample does not turn paired observations into independent samples. These structural distinctions can determine the method before any calculation begins.
Worked Examples: Identify the Procedure and Why It Fits
Worked Example: One Population Proportion
Situation. A fictional town takes a random sample of 240 adults. Of those sampled, 142 support a proposed change to the local recycling schedule. The question is whether there is evidence that more than half of all adults in the town support the change.
State. Let \(p\) be the proportion of all adults in the town who support the change. The hypotheses are \(H_0:p=0.50\) and \(H_a:p>0.50\).
Plan with conditions. There is one random sample and one binary response, support or do not support. The parameter is one population proportion, so a one-proportion \(z\) test is the appropriate procedure. Assume the town has more than 2,400 adults, so the sample is no more than 10% of the population. Under the null, the expected support count is \(np_0=240(0.50)=120\), and the expected nonsupport count is \(n(1-p_0)=240(0.50)=120\). Both are at least 10, so the Large Counts condition is met.
Do. The sample proportion is \(\hat{p}=142/240=0.5917\), rounded. The test statistic is
For the upper-tail alternative, the \(p\)-value is approximately 0.0023, rounded. Using unrounded \(\hat{p}=142/240\) gives essentially the same result.
Conclude. Because the \(p\)-value is small, there is convincing evidence that more than half of all adults in this town support the proposed recycling schedule. The one-proportion procedure fits because the question concerns one population proportion, not a mean or a comparison of two groups.
Worked Example: Two Independent Proportions
Situation. In a fictional experiment, 100 randomly assigned users receive a new reminder and 100 different users receive the usual reminder. A user either completes an online form within a week or does not. There are 68 completions in the new-reminder group and 51 in the usual-reminder group. The question asks whether the completion proportions differ.
Identify the parameter and procedure. Let \(p_1\) be the completion proportion for users assigned the new reminder and \(p_2\) the completion proportion for users assigned the usual reminder. The question concerns \(p_1-p_2\), so use a two-proportion \(z\) test. This is not a two-sample \(t\) procedure: completion is categorical, not quantitative. It is also not paired, because different users are in the two groups.
Check conditions for choosing the method. The users were randomly assigned, and each user contributes one binary outcome to one group. Because assignment places users in separate groups, the groups are independent under the design. For the test, the pooled proportion is
The pooled expected success counts are \(100(0.595)=59.5\) in each group, and the pooled expected failure counts are \(100(1-0.595)=40.5\) in each group. All are at least 10, satisfying the Large Counts condition for the test. The appropriate choice is therefore a two-proportion \(z\) test for a difference in completion proportions.
Worked Example: Paired Quantitative Measurements
Situation. A fictional group of 18 cyclists records a short-route completion time before and after a training plan. Each cyclist has both measurements. The question asks whether the plan changes the population mean completion time. A plot of the 18 within-cyclist differences shows no strong skew or outliers.
Identify the parameter and procedure. The response is quantitative time, measured in minutes. The two measurements from each cyclist are linked, so they are not independent samples. Define \(d=\text{after time}-\text{before time}\) for each cyclist. The parameter is \(\mu_d\), the population mean paired difference. Use a paired \(t\) procedure, which is a one-sample \(t\) procedure applied to the differences.
Check conditions for choosing the method. The 18 cyclists are assumed to be a random sample from the population of interest, and the sample is less than 10% of that population. The differences show no strong skew or outliers, as stated, so a \(t\) method for their mean is reasonable. The correct procedure is paired \(t\); treating the before and after measurements as two independent groups would ignore the matching and use the wrong structure.
The sign convention matters when interpreting a later result: a negative difference means a cyclist’s after time was lower than the before time. Defining the difference before choosing or interpreting the test prevents an otherwise avoidable direction error.
Worked Example: Two Categorical Variables in One Sample
Situation. A fictional random sample of 90 library members is classified by preferred reminder format and whether the member attended a scheduled workshop. The counts are shown below.
| Reminder format | Attended | Did not attend | Total |
|---|---|---|---|
| 30 | 15 | 45 | |
| Text | 20 | 25 | 45 |
| Total | 50 | 40 | 90 |
Identify the procedure. Both variables are categorical, and one random sample of library members was classified by both variables. The question is whether reminder format and workshop attendance are associated in the population of library members. Use a chi-square test of independence.
Check the expected counts. Under the null hypothesis of no association, each expected count is row total times column total divided by the grand total. For email and attendance, the expected count is \(45(50)/90=25\). For email and nonattendance, it is \(45(40)/90=20\). The corresponding text expected counts are \(45(50)/90=25\) and \(45(40)/90=20\). Every expected count is at least 5. If the sample is less than 10% of all library members, the 10% condition is also met. The table’s observed counts alone do not establish association; the chi-square test would assess whether their differences from expected counts are persuasive.
Common Mistakes and AP Exam Tips
- Choosing by the number of named groups alone. Two groups with yes/no outcomes call for a two-proportion \(z\) procedure; two groups with numerical measurements call for a two-sample \(t\) procedure. Name the response type and parameter.
- Treating paired data as independent. Before-and-after measurements on the same people are paired. State how each difference is defined and use paired \(t\) inference for quantitative measurements.
- Using a \(t\) procedure for categorical counts. A percentage may be written as a number, but a binary response is still categorical. For one or two population proportions, select a proportion \(z\) procedure.
- Confusing chi-square purposes. For one sample classified by two categorical variables, ask about independence or association. When comparing a categorical distribution across separate groups or populations, the purpose is homogeneity.
- Checking the wrong Large Counts condition. For a one-proportion \(z\) test, use the null value \(p_0\), not just the sample proportion. For a two-proportion \(z\) test, use the pooled proportion for the test condition. For chi-square, check expected counts in every cell.
- Giving only a procedure name. A complete selection explains why it fits: identify the response type, the number and relationship of samples, and the population parameter or relationship in question.
A concise, high-credit selection might say: “Because the same cyclists provide quantitative before-and-after measurements, I would analyze the within-cyclist differences with a paired \(t\) procedure for \(\mu_d\).” That sentence names the data structure, response type, procedure, and parameter.
Check Your Understanding
For each scenario, name the procedure family and the parameter or relationship it addresses.
- A random sample of 300 households is asked whether it composts. The question concerns the proportion of all households that compost.
- Two independently assigned groups receive different study reminders, and the outcome is whether each student submits an assignment on time. The question compares the submission proportions.
- The same 24 runners have their running times recorded on two different shoes. The question concerns the mean change in time.
- Two independent random samples of employees provide numerical estimates of their weekly commuting hours. The question compares the population means.
- A random sample of customers is classified by payment method and whether they use a digital receipt. The question asks whether those categorical variables are associated.
- In a one-proportion \(z\) test with \(n=150\) and \(p_0=0.40\), calculate both expected counts and decide whether the Large Counts condition is satisfied.