From Checking Conditions to Writing Them
In “Common Mistakes When Checking Conditions,” you practiced identifying the right condition and using the evidence accurately. The next step is to communicate that check in a complete sentence. A condition statement should do more than announce “met” or “not met”: it should name the relevant study detail and explain what that detail supports about the intended mean inference.
For a one-sample t procedure, a useful check usually addresses the Random Condition, independence (including the 10% condition when appropriate), and the Normal/Large Sample condition. Earlier tutorials explained what each condition means. Here, the focus is on writing evidence-based sentences: what happened in the study, how it compares with the requirement, and what conclusion is justified. If the study description does not provide enough information, say so instead of filling in the gap with an assumption.
A complete statement is contextual: it identifies the units or population relevant to the study. It is also appropriately cautious. Random selection supports generalizing to the population from which the sample was selected; it does not establish causation. Random assignment supports a causal comparison for the study units; it does not, by itself, justify generalizing to a wider population. These distinctions were developed in “Random Assignment Versus Random Sampling Conditions.”
A Three-Part Writing Routine
Think of each condition check as a short evidence argument. The condition is the claim to assess; the study description supplies evidence; and the final clause explains what follows for the proposed inference. Writing those parts together makes it easier for a reader or AP grader to see that the condition was actually checked.
Identify the procedure and the population mean or comparison of means. This keeps the statement tied to the question being answered.
Use the stated selection or assignment method, sample size, population size, or observed shape. Include a numerical comparison when the condition calls for one.
Say that a condition is supported, not supported, or cannot be verified from the information given. Avoid claiming more than the evidence establishes.
These parts need not become three separate sentences. A well-written sentence can include all three. For the 10% condition, for instance, show the comparison rather than only saying that the sample is “small.” For the Normal/Large Sample condition, state the actual \(n\) and explain which route applies. For a small sample, describe the relevant graph evidence; “the data look fine” does not say what was observed.
Writing Each Condition With Evidence
Random Condition. Identify whether the study used random sampling or random assignment, and name the group to which the statement applies. “The participants were randomized” is incomplete if the reader cannot tell whether they were randomly selected, randomly assigned, or both. If a study used a convenience sample, describe that accurately; do not label it random because the sample is large.
Independence and the 10% condition. Explain why observations can reasonably be treated as independent. When sampling without replacement from a finite population, compare \(n\) with \(0.10N\), where \(N\) is the population size. State the sample size, the population size, and the result of the comparison. If the population size or sampling method is not supplied, state that the 10% condition cannot be checked from the information given.
Normal/Large Sample condition. State the actual sample size. If \(n\geq30\), explain that the large-sample route supports an approximately Normal sampling distribution of \(\bar{x}\). This does not mean that individual observations are Normal. If \(n<30\), describe the evidence available about population shape, such as whether a sample graph shows strong skewness or a pronounced outlier. A graph provides evidence; it does not prove the population’s shape.
A condition statement can be positive, negative, or incomplete. “The condition is supported” means the evidence given supports using the procedure. “The condition is not supported” means the evidence points against the requirement. “The condition cannot be verified” means the needed information is absent. These are different conclusions, so choose the one that matches the study description.
Worked Examples
Worked Example: A Complete Check for a One-Sample t Interval
A fictional orchard randomly selects 24 pear trees from its 180 trees and records the mass, in kilograms, of fruit harvested from each tree. A histogram of the 24 values is roughly symmetric, with no apparent outliers. The orchard wants to estimate the mean fruit mass per tree for its trees.
State. Let \(\mu\) be the mean fruit mass, in kilograms, per tree in this orchard. We will check whether the conditions support a one-sample t interval for \(\mu\).
Plan. Check random selection, independence using the 10% condition, and the Normal/Large Sample condition. Since the sample size is below 30, use the sample graph as evidence about population shape rather than claiming that the large-sample route applies.
Do. The 24 trees were randomly selected from the orchard’s trees, so the Random Condition is supported for generalizing to the orchard’s trees. Because the selection was without replacement, check the 10% condition: \(0.10(180)=18\), and \(24>18\). The sample is more than 10% of the population, so the 10% condition is not met; the usual independence justification for the t interval is not supported by this check. For shape, \(n=24<30\), so the large-sample route does not apply. The histogram is roughly symmetric with no apparent outliers, which provides some evidence that the population shape is suitable for a t procedure, but does not prove that it is Normal.
Conclude. The random-selection evidence supports generalizing to the orchard’s trees, and the sample graph provides some support for the Normal/Large Sample condition. However, the 10% condition is not met because \(24>0.10(180)=18\). A complete response should report that limitation rather than conclude that every condition is satisfied.
Worked Example: When a Condition Cannot Be Verified
A fictional transit office takes a random sample of 42 bus trips and records each trip’s delay, in minutes. The office wants to estimate the mean delay for all bus trips on its route. The description does not give the number of trips in the population or say whether the sample was taken with or without replacement.
State. Let \(\mu\) be the mean delay, in minutes, for all bus trips on the route. We need to assess the condition evidence for a one-sample t interval.
Plan. Use the stated selection method for the Random Condition. Check whether independence and the 10% condition can be assessed from the sampling details. Use the actual sample size for the Normal/Large Sample condition.
Do. The 42 trips were randomly sampled, supporting the Random Condition for generalizing to trips on this route. Since \(n=42\geq30\), the large-sample route supports an approximately Normal sampling distribution of \(\bar{x}\); this does not assert that individual trip delays are Normal. Independence cannot be fully verified from the description. The population size and whether the sample was drawn without replacement are not provided, so there is no basis for checking \(n\leq0.10N\) or describing the sampling mechanism further.
Conclude. The Random Condition and the large-sample route are supported by the stated evidence. The independence check, including the 10% condition if sampling was without replacement, cannot be verified from the information provided. A careful answer identifies that missing information instead of silently assuming the condition holds.
Worked Example: Writing Checks for a Paired t Procedure
A fictional school randomly selects 15 students to try a new study schedule. Each student takes a practice quiz before and after using the schedule, and the school records each student’s change in score. A dotplot of the 15 changes is roughly symmetric, with no apparent outliers. The school wants to estimate the mean change in score for students like those in its student population.
State. Let \(\mu_d\) be the mean change in quiz score, in points, for students in the population represented by the school’s sampling process. The data are paired, so the proposed procedure is a one-sample t procedure for the 15 student-level differences.
Plan. Check randomness, independence between the students’ differences, and the Normal/Large Sample condition for the differences. For paired inference, the observations used in the procedure are the differences, not the original before and after scores treated as separate samples.
Do. The 15 students were randomly selected, supporting the Random Condition for generalizing to the school’s students. The 15 differences come from 15 distinct students, so independence between differences is plausible if the student selection does not link one student’s result to another’s. If the students were sampled without replacement from the school, the 10% condition should be checked using the school’s student population: the description does not give that population size, so the comparison cannot be completed. Since \(n=15<30\), the large-sample route does not apply. The dotplot of the differences is roughly symmetric with no apparent outliers, providing evidence that the shape condition is reasonable for a t procedure.
Conclude. The random-selection and shape evidence support the proposed paired t procedure, and the differences are the correct observations for the analysis. The independence check is plausible but the 10% comparison cannot be verified without the school’s population size. A complete condition statement makes that limitation explicit rather than treating the before-and-after measurements as independent groups.
Worked Example: Separate Checks for Two Independent Groups
A fictional recreation program randomly selects 32 participants from each of two separate, large membership lists: one list for morning classes and one for evening classes. It records the number of minutes each participant spends exercising in a typical session. The program wants to compare the mean session times for the two groups. Both sample sizes are at least 30.
State. Let \(\mu_M\) and \(\mu_E\) be the mean exercise-session times, in minutes, for morning-class and evening-class participants, respectively. We will check the conditions for a two-sample t procedure for \(\mu_M-\mu_E\).
Plan. Check random selection and independence within each group, independence between the groups, and the Normal/Large Sample condition separately for each sample. Because the lists are described as large but their sizes are not given, state what can and cannot be concluded about the 10% condition.
Do. Participants were randomly selected from each membership list, supporting the Random Condition for generalizing to the corresponding group of members. The description says the lists are large, but does not give their exact sizes or the sampling details; therefore, the 10% comparisons for sampling without replacement cannot be shown. The groups are independent because participants were selected from separate membership lists and no participant is described as appearing in both groups or being deliberately matched across them. Each sample has \(n=32\geq30\), so the large-sample route supports an approximately Normal sampling distribution for each sample mean. This does not mean that the individual session times in either group are Normal.
Conclude. The stated random selection, separate groups, and sample sizes support the Random Condition, independence between groups, and the large-sample route for both means. The within-group independence and 10% checks are not fully documented because exact population sizes and sampling details are missing. A complete response checks both samples and the relationship between groups; it does not use one group’s evidence as a substitute for the other’s.
Common Mistakes and What Full Credit Sounds Like
- Writing only “random.” Say whether units were randomly selected or randomly assigned, and identify the population or study units to which the claim applies.
- Writing “the sample is less than 10%” without showing evidence. Give \(n\), \(N\), and the comparison \(n\leq0.10N\). If \(N\) is missing, say the check cannot be completed.
- Using \(n\geq30\) to claim the data are Normal. State that the large-sample route supports an approximately Normal sampling distribution of \(\bar{x}\). Do not make a claim about the shape of individual observations based on this rule.
- Writing “the conditions are met” when a check is missing. Separate what is supported from what is unknown. Missing population size, sampling details, or graph evidence should be identified directly.
- Checking the wrong observations for paired data. For a paired t procedure, check the distribution of the within-pair differences. For two-sample t procedures, assess shape in each group and independence between groups separately.
- Overstating what a sample graph proves. Use wording such as “the sample graph is roughly symmetric with no apparent outliers, providing support for…” rather than “the population is definitely Normal.”
For full-credit communication, connect evidence to the condition and to the scope of the inference. “Random sample” is evidence; “therefore, generalizing to the population represented by that sample is supported” explains its relevance. “\(n=32\)” is a fact; “because \(32\geq30\), the large-sample route supports an approximately Normal sampling distribution of \(\bar{x}\)” explains what it means. When evidence is absent, a direct statement that the condition cannot be verified is more accurate than an unsupported yes or no.
Check Your Understanding
For each situation, write a complete contextual condition statement or identify what information is missing.
- A random sample of 19 people is taken without replacement from a club of 250 members. Check the 10% condition in a sentence.
- A sample has \(n=34\), and no graph of individual observations is provided. Write a correct Normal/Large Sample condition statement without claiming that individual data are Normal.
- A researcher says that 12 measurements were “collected from volunteers.” What can you say about the Random Condition, and what would you need to know to make a population claim?
- In a paired study with 17 people, the graph of the before-and-after differences is strongly right-skewed with a pronounced outlier. Identify which data should be used for the shape check and what the evidence suggests.
- Two groups have sample sizes of 31 and 36, but the study description does not say whether people appear in both groups. What independence evidence is missing, and why should the two sample sizes not replace that check?