A Failed Condition Is a Reason to Pause
A calculator can produce a t interval or test even when the conditions for mean inference are not supported. The calculation alone does not show that the result is trustworthy. In “Verifying Conditions From Summary Statistics Only,” you learned to separate evidence that supports a condition from information that is missing. Now we consider what to do when an audit finds a condition is not met—or cannot be checked.
The right response depends on the condition. A small sample with a pronounced outlier raises a different problem from a convenience sample or observations that are linked. Sometimes more information can resolve uncertainty. Sometimes better data must be collected. Sometimes the available data still describe the observed units, but a population inference is not justified.
As explained in “Why Conditions Matter in Mean Inference,” randomness, independence, and an appropriate shape support different parts of an inference. A response should target the specific problem. Collecting a larger sample may help with the Normal/Large Sample condition, but it does not create random selection or remove links among observations.
Match the Response to the Problem
The table gives a practical starting point for a one-sample, paired, or two-sample t procedure. For paired and two-sample procedures, assess the relevant observations separately, as described in “Conditions for One-Sample Versus Paired Data” and “Conditions for Two-Sample t Procedures.”
| Condition problem | What it affects | Reasonable response |
|---|---|---|
| Random selection or an appropriate randomized process is absent | Whether the result supports a claim about a population or study units beyond those observed | Limit the conclusion to the data collected, or collect data using an appropriate chance process. A larger convenience sample does not fix selection bias. |
| Independence is doubtful, or the 10% condition fails | The usual standard error and t model may not describe the sampling variability correctly | Investigate links, repeated measurements, clustering, and the sampling fraction. Collect data under a design that supports independence, or use a method suited to the actual design rather than pretending the observations are independent. |
| Small sample with strong skewness or a pronounced outlier | The t procedure may be unreliable because the data provide weak support for the t model | Check for a data-entry or measurement error, examine the study design, and consider collecting more appropriate data. Do not automatically delete an unusual value or rely on the t procedure without qualification. |
| A condition cannot be checked from the report | The evidence is insufficient to justify the intended inference | Request the missing design details, population size, or data display. State that the condition is unclear until evidence is available. |
A response can also be to narrow the claim. A nonrandom sample may still describe the people or items actually observed, but it does not automatically represent a broader population. If the goal is to describe every member of a finite group and measurements are available for all of them, report that group’s mean as a descriptive statistic; do not present it as an estimate from a sample.
Sometimes a different t procedure is appropriate, but only when the design supports it. For example, if two measurements are genuinely paired on the same individuals, analyze the within-person differences with a paired t procedure rather than treating the measurements as independent groups. A procedure cannot be selected just because it appears to avoid a condition problem; it must match how the data were collected.
When Shape Evidence Is the Problem
For mean inference, the Normal/Large Sample condition is supported if the population is approximately Normal or if the large-sample route applies. “Using the n at Least 30 Rule Correctly” explains the role of \(n\geq30\). For a smaller sample, the graphs and observations matter. “Robustness of t Procedures” also explains why the t procedure can tolerate some departures from Normality, but is less dependable with a small sample that has strong skewness or pronounced outliers.
If the sample is small and the shape evidence is poor, do not describe the t procedure as valid merely because it is often robust. First check whether unusual values are errors. If a value is a genuine observation, deleting it simply to make a graph look more Normal is not justified. More data may provide better evidence about the population and can make the large-sample route available, but a larger \(n\) does not guarantee that every problem disappears. A highly unusual population shape or extreme values still call for caution.
If the shape condition is unclear rather than clearly poor, request the relevant display—such as a dotplot, histogram, boxplot, or Normal probability plot—when the sample is small. If no such information can be obtained, state that the shape condition cannot be verified and avoid claiming that the t-based inference is fully justified.
A Decision Workflow
Identify the population mean or means, or the mean of paired differences. Keep the target and the data-collection design in view.
Decide whether randomness, independence, the 10% condition, or the Normal/Large Sample condition is not met or is simply unclear. Use evidence from the study, not guesses.
Request missing details, check a graph, investigate possible errors, limit the claim, or collect new data using a suitable design. Do not assume one fix repairs every condition.
Say what the evidence does and does not justify. If a key condition is not supported, do not present the usual t interval or test as a fully justified population inference.
The response is not always “collect a bigger sample.” It might be “collect a random sample,” “find out whether these measurements are linked,” or “get a graph of the individual observations.” The next examples show how to connect the condition evidence to a practical next step.
Worked Examples
Worked Example: A Small Sample With a Pronounced Outlier
A fictional environmental club records the number of minutes it takes 11 sampled volunteers to sort a box of recyclable materials. The sample was randomly selected from the club’s 48 volunteers. A boxplot shows ten times between 8 and 16 minutes and one time of 41 minutes. The club wants to use a one-sample t interval to estimate the mean sorting time for all 48 volunteers.
State. Let \(\mu\) be the mean sorting time, in minutes, for the 48 volunteers. We need to decide whether a one-sample t interval is justified and what action is appropriate.
Plan. Audit randomness, independence, the 10% condition, and the Normal/Large Sample condition. The sample is small, so the observed shape is important. If a condition is not supported, identify whether more information, a different design, or a more limited conclusion is appropriate.
Do. The volunteers were randomly selected, supporting the Random Condition for this club. Since the sample was taken without replacement, check \(n\leq0.10N\): \(11\leq0.10(48)=4.8\) is false. The sample is more than 10% of the club, so the usual independence justification from the 10% condition is not supported. Also, \(n=11<30\), and the boxplot shows a pronounced high value far from the other observations. The Normal/Large Sample condition is therefore not supported by the evidence given. We should check the original record to determine whether 41 minutes is a recording or measurement error, but we should not discard it if it is a genuine sorting time.
Conclude. These data do not justify the usual one-sample t interval for \(\mu\): the 10% condition fails, and the small sample has a pronounced outlier. The club could report the observed times descriptively, verify the unusual value, and consider collecting data under a design appropriate to the target. It should not claim that a routine t interval has reliable coverage for the mean based on these checks.
Worked Example: A Large Convenience Sample Does Not Fix Selection
A fictional school technology committee asks 52 students who are waiting near the library whether they support extending computer-lab hours. Their responses are recorded on a five-point scale, and the committee calculates the sample mean. It wants to use a one-sample t interval to estimate the mean opinion score of all students at the school.
State. Let \(\mu\) be the mean opinion score, on the five-point scale, for all students at the school. We need to assess whether the sample supports this population inference.
Plan. Consider how students entered the sample, then assess the other conditions for a one-sample t procedure. A sample size of at least 30 supports the large-sample route for shape, but it cannot establish random selection or independence.
Do. The students were a convenience sample rather than a random sample, so the Random Condition is not met for generalizing to all students. Students near the library at that time may differ from other students in ways related to computer-lab use. Since \(n=52\geq30\), the large-sample route supports the Normal/Large Sample condition. But that does not remove the selection problem. The description also does not establish whether the students’ responses can be treated as independent; we would need to know whether each student was counted once and whether the sampling process introduced links. If selection was without replacement from the school population, we would need the population size to check the 10% condition.
Conclude. The committee should not use this convenience sample’s t interval as a justified estimate of the mean opinion score for all students. It can describe the 52 respondents, while clearly identifying them as a convenience sample. To make a population inference, it should collect a random sample from a defined list of students and check independence and the remaining conditions. Simply asking more students in the same location would not fix the selection method.
Worked Example: A Sample Is Too Large a Fraction of the Population
A fictional museum has 360 annual members and randomly selects 60 members without replacement to estimate the mean number of visits per member during the past year. The visit counts show no unusual shape, and each selected member contributes one observation. The museum proposes a one-sample t interval.
State. Let \(\mu\) be the mean number of visits during the past year for the museum’s 360 annual members. We need to determine whether the usual independence check supports a one-sample t interval and what the museum could do next.
Plan. Confirm random selection and one observation per member. For sampling without replacement, check the 10% condition by comparing \(n\) with \(0.10N\). Assess the shape condition separately; favorable shape evidence does not resolve an independence problem.
Do. The random selection supports the Random Condition, and each member contributes one observation. But \(n=60\), while \(0.10N=0.10(360)=36\). Since \(60>36\), the sample exceeds 10% of the population, so the usual 10% condition is not met. The stated shape evidence is favorable, but that does not repair the independence concern. The museum could instead select a random sample of no more than 36 members if it wants to use this standard 10% check, or measure all 360 members and report the mean for that group descriptively. If it uses a sampling design that does not meet the usual condition, it should not present the ordinary t interval as though the condition had been satisfied.
Conclude. The sample is random, but the 10% condition fails because 60 is more than 10% of 360. The museum should change its data-collection plan or be explicit that the usual one-sample t interval is not justified by the standard independence check. The favorable shape information alone is not enough to proceed without qualification.
Worked Example: Missing Information About a Small Sample
A fictional bike-repair shop receives a report with \(n=15\), a sample mean repair time of 38 minutes, and a sample standard deviation of 9 minutes. The report does not say how the repairs were selected or provide a graph. The shop wants an interval for the mean repair time of all repairs completed that month.
State. Let \(\mu\) be the mean repair time, in minutes, for all repairs completed by the shop that month. We need to decide what the available information permits.
Plan. Use the sample size to assess the large-sample route, but do not infer the sampling method or data shape from the mean and standard deviation. Identify the specific information needed before deciding whether a one-sample t interval is appropriate.
Do. Since \(15<30\), the large-sample route is not available. The report gives no graph or information that the repair-time distribution is approximately Normal, so the Normal/Large Sample condition is unclear. The reported mean and standard deviation cannot show whether times are skewed or include outliers. The selection method is also absent, so randomness is unclear. If the repairs were sampled without replacement from a finite set, the shop would need the size of that set to check the 10% condition; it would also need to confirm that each repair contributes one independent observation.
Conclude. The report does not provide enough evidence to justify a t interval for the mean repair time. The shop should request the selection details and, because the sample is small, a graph of the 15 repair times or information about the population’s shape. Until then, it should describe the interval as unsupported by the available condition checks rather than treating missing evidence as proof that the conditions are met.
Common Mistakes and AP Exam Tips
- Using a larger sample as a cure-all. A larger sample may support the large-sample route for shape, but it does not make a convenience sample random or unlink observations that are dependent.
- Calling a condition “met” when the evidence is missing. Use “unclear” and name what evidence is needed. For example: “The report does not describe how repairs were selected, so the Random Condition cannot be verified.”
- Deleting a genuine outlier to make the procedure work. Investigate possible recording errors, but retain valid observations. If the value is real, discuss how it affects the shape evidence instead of removing it without justification.
- Assuming a different procedure automatically solves the problem. A paired t procedure is appropriate only for genuine pairs and their differences. A new method must match the study design and address the actual condition problem.
- Claiming too much from a nonrandom sample. A t interval does not turn a convenience sample into a representative sample. Limit the description to the sampled respondents or collect suitable random data.
- Ignoring a failed condition because the graph looks reasonable. Conditions address different concerns. Evidence about shape cannot make a sampling fraction satisfy the 10% condition.
A strong AP response identifies the exact evidence, explains the consequence, and proposes a relevant next step. For example: “Because \(n=15<30\) and no graph or population-shape information is provided, the Normal/Large Sample condition is unclear. We need a graph of the repair times or information about the population distribution before deciding whether the t procedure is appropriate.” Avoid simply writing “conditions fail” without specifying which one and why.
Check Your Understanding
For each situation, identify the condition problem and describe an appropriate response.
- A random sample of 14 water samples has a strong right-skewed distribution and one unusually high value. What should be checked before deciding whether a t procedure is appropriate?
- A survey uses 80 volunteers who respond to a school website announcement. Would increasing the number of volunteers necessarily justify inference to all students? Explain.
- A random sample of 45 items is taken without replacement from a batch of 300 items. Does the 10% condition hold? Show the comparison and state a possible response.
- A report gives \(n=12\), \(\bar{x}\), and \(s\), but no study description or graph. Name two kinds of information needed before assessing a t procedure.
- Two groups’ measurements are linked because each person appears in both groups. What feature of the design should be clarified before selecting a mean-inference procedure?