Read the Design Before You Choose a Procedure
A study description rarely presents its conditions in a tidy checklist. It may say how participants were recruited but omit the population size, or report the sample size without showing the data’s shape. Your task is to sort the available evidence: which conditions are supported, which are not met, and which cannot be decided from the description?
In “Why Conditions Matter in Mean Inference” and the tutorials on checking randomness, independence, and the Normal/Large Sample condition, we examined the conditions themselves. Here, the new skill is reading a study description closely enough to make a careful judgment without filling in missing facts. A calculator cannot supply evidence the study description does not give.
A Condition Evidence Audit
Use a condition evidence audit: list each condition relevant to the intended mean procedure, locate the sentence or fact that bears on it, and assign a status. For a one-sample t procedure, the main checks are randomness, independence (including the 10% condition when appropriate), and the Normal/Large Sample condition. For a two-sample t procedure, assess independence within each group and between groups, as well as the shape condition for each group.
Determine what one observation represents, whether the study uses one sample or two groups, and how the study units were selected or assigned.
Look for an explicit chance process, such as a random sample or random assignment. Do not treat “participants were recruited” as equivalent to either one.
Check for repeated or linked observations, clustering, and—when sampling without replacement from a finite population—whether the sample is no more than 10% of that population.
Use the population description, sample size, and any stated graphs or distribution features to assess the Normal/Large Sample condition. Do not invent a graph or assume a population shape.
For each condition, say what supports it, what contradicts it, or what information is missing. Then state whether the described evidence is enough to justify proceeding.
The statuses should reflect the evidence, not your guess about what probably happened. “The study says participants were randomly selected” supports the Random Condition. “The study does not state whether selection was random” leaves that condition unclear. “Participants volunteered” indicates that random selection was not used; it does not support generalizing to a population through random sampling.
For sampling without replacement, the 10% condition compares the sample size \(n\) with the population size \(N\): check whether \(n\leq0.10N\), equivalently \(N\geq10n\). If \(N\) is not given and cannot be determined, mark that check unclear rather than silently assuming the population is large. The 10% check is not a substitute for investigating other sources of dependence, such as repeated measurements on the same person.
For the Normal/Large Sample condition, the earlier tutorial “Checking the Normal/Large Sample Condition” established the main routes: an approximately Normal population or a sample size of at least 30. With fewer than 30 observations, look for relevant evidence about the data’s shape, such as a description of a graph. If the study description supplies no such evidence, the shape check is unclear; the fact that a calculator can run a t procedure does not resolve it.
Worked Examples
Worked Example: A Small Random Sample With Shape Evidence
A fictional veterinary team randomly selects 20 dogs from a registry of 500 dogs enrolled in a wellness program. It records each dog’s recovery time, in days, after a routine procedure. The report says that a dotplot of the 20 recovery times is roughly symmetric, has one main cluster, and shows no apparent outliers. The team plans to use a one-sample t procedure for the mean recovery time among dogs in the registry.
State. Let \(\mu\) be the mean recovery time, in days, for dogs represented by the registry. We will audit the conditions for a one-sample t procedure using only the study details provided.
Plan. Check the Random Condition, independence and the 10% condition, and the Normal/Large Sample condition. Because this is a sample taken without replacement from a stated finite population, compare \(n\) with \(0.10N\). Since \(n=20\), the large-sample route does not apply; use the reported dotplot description for the shape check.
Do. The team randomly selected the dogs, so the Random Condition is supported for generalizing to dogs represented by the registry. The sample is 20 from 500, and \(0.10(500)=50\). Since \(20\leq50\), the 10% condition is met, supporting independence for sampling without replacement. The report describes a roughly symmetric dotplot with one main cluster and no apparent outliers, which supports the Normal/Large Sample condition for this small sample.
Conclude. The description supports all three conditions for the one-sample t procedure. Based on the information given, using that procedure to make an inference about the mean recovery time for dogs represented by the registry is reasonable. The conclusion should not automatically extend to dogs outside the registry.
Worked Example: Randomness and Independence Supported, Shape Unclear
A fictional municipal lab randomly selects 25 water samples from a list of 400 independently scheduled household sampling visits. It measures the lead concentration, in micrograms per liter, in each sample. The study description gives no graph, summary of the distribution, or information about the population’s shape.
State. Let \(\mu\) be the mean lead concentration for the population represented by the sampling list. We need to determine which conditions for a one-sample t procedure are supported and which remain uncertain.
Plan. Check whether selection was random, whether the 10% condition is satisfied for sampling without replacement, and whether the available information supports the Normal/Large Sample condition. In particular, \(n=25\) is below 30, so the sample size alone does not establish the large-sample route.
Do. Random selection is stated, so the Random Condition is supported. For independence, \(0.10(400)=40\), and \(25\leq40\), so the 10% condition is met. The description also says the visits were independently scheduled, with one measurement from each visit, and gives no indication that the observations are repeated or linked. However, \(n=25<30\), and the description gives no sample graph or population-shape information. Therefore, the Normal/Large Sample condition is unclear.
Conclude. The study description supports randomness and independence, including the 10% check, but it does not provide enough evidence to assess the shape condition for this sample of 25. We should not claim that all conditions are met. We would need a suitable graph or information about the population distribution before deciding whether the one-sample t procedure is justified.
Worked Example: A Randomized Experiment With Two Small Groups
A fictional ergonomics team recruits 36 volunteers and uses a chance process to assign 18 to a new keyboard setup and 18 to their usual setup. Each participant uses only the assigned setup for one week, and the team records one typing-speed measurement per participant. The report gives no graphs or description of the typing-speed distributions in either group. The team plans to compare the two population means for the study participants.
State. Let \(\mu_N\) be the mean typing speed for participants assigned to the new setup and \(\mu_U\) the mean for participants assigned to their usual setup. We will check the two-sample t conditions using the experiment’s description.
Plan. For two-sample t inference, assess the randomized design, independence within each group and between groups, and the Normal/Large Sample condition separately for each group. The volunteers were assigned to treatments rather than randomly sampled from a wider population, so assignment and sampling must not be confused.
Do. The chance process assigned participants to the two setups, supporting the Random Condition for a causal comparison among these study participants. Each participant appears in only one group and contributes one measurement; the description does not indicate repeated or linked observations. The two groups were formed by random assignment, not by matching participants or reusing the same participants, which supports treating them as independent groups for this comparison. Because assignment was used rather than sampling without replacement from a stated finite population, a 10% check is not the relevant justification here. Each group has \(n=18<30\), and the description provides no graphs or distribution information. The Normal/Large Sample condition for each group is therefore unclear.
Conclude. The design supports a randomized comparison, and the description gives evidence for independent observations and groups. But the shape condition for each small group cannot be assessed from the information provided. We cannot say all conditions for the two-sample t procedure are verified. Also, because the participants volunteered rather than being randomly selected from a broader population, the design does not by itself support generalizing the result to all people who use keyboards.
Worked Example: A Large Convenience Sample Does Not Fix Randomness
A fictional student wellness club invites people who visit its booth to volunteer for a survey about daily walking time. Forty-two volunteers report their walking time, and the club plans to use a one-sample t procedure to estimate the mean for all students at the school. The report describes a roughly symmetric histogram with no apparent outliers. It does not state how many students attend the school.
State. Let \(\mu\) be the mean daily walking time for all students at the school. We will decide what the description supports about the conditions and about the proposed population inference.
Plan. Check the recruitment method, any relevant independence evidence, and the shape condition. Do not treat a large sample or an acceptable-looking histogram as evidence that volunteers were randomly selected. The 10% condition also requires a sample drawn without replacement from a finite population with an appropriate population count.
Do. The students volunteered at a booth; no random selection is described. Thus, random sampling from all students is not supported, and the volunteers may differ from students who did not visit or respond. The description does not say whether observations are linked or whether the 10% condition applies, so those independence details are unclear. Since \(n=42\geq30\), the large-sample route supports the Normal/Large Sample condition under the usual AP check. The histogram description is also consistent with that condition, but neither fact repairs the lack of random selection.
Conclude. The shape condition is supported, but the description does not justify generalizing the mean from these volunteers to all students at the school. Some independence details are also missing. A large sample can help with the sampling distribution’s shape; it cannot turn a volunteer sample into a random sample.
Common Mistakes and AP Exam Tips
- Calling missing information a failure. If the study never says whether the population is approximately Normal and the sample is small, write “unclear from the description,” not “the condition is violated.”
- Calling missing information evidence. Do not assume a study used random selection because it has many participants, or assume a population is Normal because the study reports a mean.
- Mixing up selection and assignment. Random assignment supports a randomized comparison; it does not mean participants were randomly sampled from a population. State what the chance process did.
- Checking the 10% condition without a population size. For sampling without replacement, show the comparison when \(N\) is known. If \(N\) is missing, say the 10% condition cannot be checked from the description.
- Using one group’s shape evidence for both groups. For two-sample t inference, assess the Normal/Large Sample condition separately for each group. A graph or sample size in one group does not automatically settle the other group’s check.
- Letting one good condition excuse another weak one. A random sample does not establish an appropriate shape, and \(n\geq30\) does not establish random selection. Report each condition on its own evidence.
A full-credit conditions statement is specific and calibrated: “The study randomly selected 25 visits from 400, so the Random Condition is supported and \(25\leq0.10(400)=40\), meeting the 10% condition. Because \(n=25<30\) and no graph or population-shape information is reported, the Normal/Large Sample condition is unclear.” This explains both the judgment and its basis without claiming more than the description provides.
Check Your Understanding
For each situation, identify what is supported, what is not met, and what remains unclear for the relevant mean inference.
- A random sample of 28 residents is selected without replacement from a town of 1,200. No graph or information about the population’s shape is reported. Which conditions can you assess, and what remains unclear?
- A study says that 35 participants volunteered and were randomly assigned to two treatments. Does that establish random sampling from the town where the study took place?
- A random sample of 45 devices is selected from a known inventory of 300. Check the 10% condition and state what the sample size tells you about the Normal/Large Sample condition.
- In a two-group study, one group has 32 observations and the other has 19. The description gives a graph only for the smaller group. What must you check separately?
- Write one sentence that distinguishes “the condition is unclear” from “the condition is not met.”