Look for How the Groups Were Formed
In “Spotting Paired Designs in Word Problems,” you learned to look for a design-based link between two observations. Now look for the other common structure in a mean comparison: two groups with no one-to-one links between their observations. Such groups may come from two separate random samples or from randomly assigning study units to two groups.
The wording can be subtle. A problem may give two lists, two columns, or two sample sizes, but the layout does not tell you how the data were collected. Instead, ask: Were different units selected for two groups, or were units assigned by chance to one of two groups? Is there any design-based reason to match one observation in the first group with a particular observation in the second?
Here, “independent” describes the structure of the study, not a conclusion you can prove just by inspecting the numbers. Different people, objects, or other units are in the two groups, and the design does not link one unit to a particular unit in the other group. You still need to consider whether observations within each group are independent, as well as whether the groups themselves were formed in a way that supports inference.
As in “Identifying the Parameter in a Mean Problem,” define the response and the population means before naming a procedure. If group 1 and group 2 represent two populations, the target is usually \(\mu_1-\mu_2\), the difference in their true mean responses. The order matters: group 1 minus group 2 determines how to interpret a positive or negative difference.
Two Routes to Independent Groups
Separate random samples: A researcher selects a random sample from one population and a second random sample from another population. For example, the researcher might randomly select clinics from an urban-clinic roster and independently select clinics from a rural-clinic roster. The samples consist of different clinics, and no clinic in one sample is deliberately matched to a clinic in the other.
The word “independently” matters when describing the sampling. Selecting one group and then choosing the other to match its members would create a different design. Also check whether the populations are distinct in the context: a clinic should not be eligible to appear in both the urban and rural rosters if the study intends two separate groups.
Random assignment: In an experiment, a researcher uses chance to assign study units to one of two treatment groups. Each unit receives one treatment and contributes a response to that group. If the researcher does not deliberately match units or measure each unit under both treatments, the groups are unpaired.
Random sampling and random assignment serve different purposes. Random sampling can support generalizing findings to the population represented by the sampling frame. Random assignment can support a cause-and-effect conclusion about the treatments, provided the experiment is well designed. Random assignment by itself does not make a volunteer sample representative of a larger population.
Clues That Do Not Establish Pairing
A frequent source of confusion is mistaking a convenient arrangement for a genuine link. Equal sample sizes, similar group averages, matching row numbers, or data stored side by side do not establish pairs. A pair requires a meaningful connection created by the study design, such as the same unit measured twice or a deliberate matching rule.
Similarly, two groups can be comparable without being paired. Random assignment aims to create treatment groups that are comparable overall, but it does not match each participant in one group to a particular participant in the other. With ordinary individual random assignment, each participant receives one treatment; the comparison is between groups, not between participant-by-participant differences.
A useful check is to cover the numbers and ask whether the study description still tells you which observation in group 1 belongs with which observation in group 2. If it does not, there is no described one-to-one pairing. Then ask whether random samples or random assignment support the comparison. The answer to the first question identifies the structure; the second helps assess what inference is justified.
Worked Example: Two Random Samples of Clinics
Worked Example: Clinic Wait Times in Two Regions
A health-planning team independently selects 40 urban clinics from a roster of 600 urban clinics and 35 rural clinics from a separate roster of 500 rural clinics. At each selected clinic, the team records the mean patient wait time on one specified day, in minutes. The urban sample mean is 31 minutes, and the rural sample mean is 36 minutes.
Identify the design: The team took two separate random samples. Each sampled clinic is in one regional group, and the description gives no rule pairing an urban clinic with a particular rural clinic. These are independent, unpaired samples for comparing mean wait times.
Check the sampling fractions: For the urban sample, \(40/600=0.0667\), or about 6.7%. For the rural sample, \(35/500=0.0700\), or 7.0%. Each sample is less than 10% of its population, so the 10% condition is satisfied for sampling without replacement.
Identify the target: Let \(\mu_U\) be the true mean wait time, in minutes, for urban clinics in the population represented by the urban roster, and let \(\mu_R\) be the corresponding mean for rural clinics. The target is \(\mu_U-\mu_R\). The observed sample difference is \(31-36=-5\) minutes, but that sample statistic is not itself the difference in population means.
Explain the procedure choice: If the team wants to make an inference about the difference in mean wait times, the two-sample t procedure is the candidate procedure, subject to its other conditions. As covered in “Conditions for a Two-Sample t Test,” the team must also consider independence of observations within each group and whether each group’s data are reasonably compatible with t inference. The random samples support generalizing to the populations represented by the rosters, not automatically to every clinic in every region.
Worked Example: Random Assignment to Two Treatments
Worked Example: Two Study Routines
A researcher recruits 50 volunteers for a study of study routines. Using a random process, the researcher assigns 25 volunteers to a flashcard routine and 25 to a practice-question routine. After two weeks, each volunteer completes the same assessment once. The response is the number of questions answered correctly. The sample means are 18.4 for flashcards and 20.1 for practice questions.
Identify the design: This is a randomized experiment with two unpaired treatment groups. Each volunteer is assigned to only one routine and contributes one assessment score. Random assignment does not mean that the first flashcard participant is matched to the first practice-question participant; no such pairing is described.
Identify the target: Let \(\mu_F\) be the true mean assessment score for the population of interest under the flashcard routine, and let \(\mu_P\) be the corresponding mean under the practice-question routine. Define the difference as \(\mu_F-\mu_P\), measured in correctly answered questions. The observed difference is \(18.4-20.1=-1.7\) questions.
Explain what random assignment supports: If the experiment is carried out appropriately, random assignment supports using the comparison to assess whether the routines cause a difference in mean scores for the study participants or a population to which the experiment reasonably applies. The volunteers were not described as a random sample of all students, so random assignment alone does not justify generalizing to all students.
Name the candidate procedure: For inference about a difference in mean scores, the two-sample t procedure is appropriate if its conditions are met. The groups are independent in the design because each volunteer is in one group only and the study does not create matches. The researcher should still consider whether responses within each group are independent and whether the score distributions are suitable for t inference.
Worked Example: Unpaired Groups Without Random Selection
Worked Example: Two Existing Class Sections
A teacher compares quiz scores for two existing class sections. Section A has 27 students and uses a new review activity; Section B has 29 different students and uses the usual review. The teacher chose which section would use the new activity, and neither section was randomly selected from a larger student population. The response is the quiz score, in points.
Identify the structure: These are two unpaired groups because they contain different students, each student receives one review approach, and no students are deliberately matched. The unequal group sizes, 27 and 29, do not affect that classification.
Separate structure from inference: The groups are unpaired, but the description does not provide random sampling or random assignment. The groups may be compared descriptively, for example by reporting each section’s mean score, but the design does not provide the usual random basis for a two-sample t inference. A difference between the section means could reflect other differences between the sections, not just the review activity.
State what is missing: Let \(\mu_A\) and \(\mu_B\) represent the true mean quiz scores for clearly defined populations corresponding to the two teaching approaches. To use a two-sample t procedure for a broader inference, the study would need an appropriate random design and the remaining conditions. Merely having 27 and 29 observations does not supply that design.
Avoid an overclaim: Calling the sections “independent samples” does not turn a nonrandom comparison into a randomized experiment or a random sample. The groups are structurally unpaired; whether inference is justified is a separate question.
A Decision Routine for Word Problems
Before selecting a mean-inference procedure, use the study description to classify the groups. This builds on “Paired t Versus Two-Sample t” and “Spotting Paired Designs in Word Problems,” but focuses on the evidence that groups are unpaired and on the source of randomization.
Confirm that the comparison concerns a quantitative measurement, such as minutes, score points, or grams.
Check whether the groups contain different people, objects, or other units, with each unit contributing to one group only.
Look for repeated measurements on the same unit or deliberate one-to-one matching. If neither is present, do not invent pairs based on list order or equal sizes.
Decide whether the groups came from two separate random samples, random assignment to treatments, or neither.
For unpaired groups, define \(\mu_1\) and \(\mu_2\) for the relevant populations or treatment conditions, and state the order of \(\mu_1-\mu_2\).
Unpaired structure points toward two-sample t inference for means, but random design, independence, the 10% condition when relevant, and distributional conditions still need attention.
Common Mistakes and AP Exam Tips
- Calling equal-sized groups paired: Equal sample sizes do not identify partners. Full-credit reasoning explains whether the same units were measured twice or a deliberate matching rule linked units.
- Pairing by row or order: Spreadsheet layout is not a study design. Unless the problem says which observations are matched, do not calculate or analyze row-by-row differences.
- Confusing random assignment with random sampling: Assignment to treatments can support a cause-and-effect conclusion, while random sampling can support generalization to a represented population. State the process the problem actually describes.
- Assuming unpaired means inference is automatically valid: Unpaired describes the relationship between groups. It does not by itself establish randomization, independence within groups, the 10% condition, or suitable group distributions.
- Claiming random assignment creates matched pairs: Ordinary assignment to two groups places each unit in one group; it does not create a particular partner in the other group. State that the groups are unpaired unless a matching design is explicitly described.
- Leaving the parameter vague: Define the two population means and the subtraction order. For example, \(\mu_1-\mu_2\) is the true mean for group 1 minus the true mean for group 2, in the response’s units.
- Overstating the conclusion a design permits: A nonrandom comparison of existing groups may describe those groups, but it does not have the same inference basis as random samples or a randomized experiment.
A concise AP response might say: “The study uses two separate random samples of different clinics, with no design-based matching, so the groups are unpaired. Let \(\mu_1\) and \(\mu_2\) be the population mean wait times for the two clinic populations; the target is \(\mu_1-\mu_2\).” For an experiment, name random assignment rather than random sampling and explain its role accurately.
Check Your Understanding
For each situation, decide whether the groups are unpaired and identify the random process, if any. State what comparison of means would target when appropriate.
- A city independently selects 30 north-side parks and 25 south-side parks from separate park rosters, then records litter mass in kilograms at each park. Are the groups paired? What supports generalizing to the roster populations?
- A researcher assigns 40 volunteers by chance to one of two sleep schedules. Each person follows only one schedule and completes one attention assessment. Are the groups paired? Does random assignment establish a random sample of all adults?
- Two lists each contain 18 delivery times. The first value in each list was recorded on the same spreadsheet row, but the drivers were not matched and no driver was measured under both conditions. Is there a basis for pairing the rows?
- A principal compares test scores from two existing classes, with 24 students in one and 26 in the other. No students were randomly assigned or randomly sampled. Are the groups unpaired, and does that alone justify a two-sample t inference?
- A researcher independently samples 50 households from each of two regions. Define a suitable parameter for comparing mean monthly electricity use, and state the units of the difference.