Two Groups Mean Two Sets of Conditions to Check
A two-sample t procedure compares the means of two quantitative groups. Unlike a paired t procedure, it uses two separate samples rather than one list of within-pair differences. As explained in “Conditions for One-Sample Versus Paired Data,” whether observations are paired depends on the study design—not simply on having two groups. For a two-sample t procedure, the groups should be independent.
There are three broad checks: randomness, independence, and Normality or a sufficiently large sample. Independence has two parts to consider: observations should be independent within each group, and the two groups should be independent of each other. The shape condition also applies separately to each group. A large sample in one group does not make a small sample in the other group large.
The Conditions for Two-Sample t Procedures
Start by identifying the two populations or treatments being compared and the observational units in each group. For example, if each observation is one randomly selected tree, then the trees—not the leaves measured on each tree—are the units. This helps make the randomness and independence checks concrete.
Random condition. Random samples support generalizing to the populations from which they were selected. In a randomized experiment, random assignment supports a comparison of the treatment groups. Random assignment does not, by itself, make the participants representative of a broader population. As in “Checking the Random Condition,” explain what was actually randomized and what that randomization supports.
Independence within each group. Consider whether the observations from different units in the same group can reasonably be treated as independent. For a random sample drawn without replacement from a finite population, use the 10% condition: the sample size must be no more than 10% of that population. Check this separately for each sample, as in “Checking the 10% Condition for Independence.” If a group is a census or comes from another design, describe the relevant design instead of claiming a 10% condition that does not apply.
Independence between groups. The two groups should come from separate, unrelated units or be assigned in a way that makes the treatment groups independent. If the same person contributes one observation to each group, or if people are deliberately matched across groups, the data are paired; use a paired t procedure rather than treating the groups as independent. Equal sample sizes do not establish independence.
Normal/Large Sample condition. Check the shape for each group separately. For a group with \(n\geq30\), the AP large-sample route is met. If a group has fewer than 30 observations, use evidence about that group’s population shape or inspect its sample graph for strong skewness or pronounced outliers. The graph-reading and robustness ideas from “Checking the Normal/Large Sample Condition” and “Robustness of t Procedures” apply to each group on its own.
A two-sample t procedure does not require the two groups to have the same sample size. It also does not require you to pool the groups when assessing Normality. If one group has a small sample, that group still needs appropriate shape evidence even when the other group has many observations.
A Practical Condition-Checking Workflow
Before calculating a two-sample t statistic or interval, organize the checks by group and by design. This prevents one reassuring feature—for example, a large combined sample—from hiding a problem in one group.
Name the two populations or treatments and identify what one observation represents in each group. Check that measurements are not naturally paired or matched.
State whether each group came from a random sample or whether treatments were randomly assigned. Explain whether the design supports generalization, a causal comparison, or both.
Consider independence among units within each sample and apply the 10% condition separately when sampling without replacement. Then explain why the two groups themselves can be treated as independent.
Use \(n\geq30\) for a group’s large-sample route. For a smaller group, assess that group’s population shape or sample graph, noting skewness and pronounced outliers.
When writing a response, make each check specific to the study. “The samples are independent” is less informative than explaining how the units were sampled, whether a 10% condition applies, and why the groups do not overlap or form pairs.
Worked Examples
Worked Example: Two Large Random Samples
A fictional environmental team randomly selects 42 trees from one large preserve and 36 trees from a second large preserve. It measures the annual growth of each tree, in centimeters. The two preserves contain more than 420 and 360 eligible trees, respectively. The team wants to compare the mean annual growth in the two preserve populations. Both sample sizes are at least 30.
State. Let \(\mu_1\) be the mean annual growth, in centimeters, of trees in the first preserve, and let \(\mu_2\) be the corresponding mean for trees in the second preserve. We are assessing whether a two-sample t procedure for \(\mu_1-\mu_2\) is appropriate.
Plan. Check random selection, independence within each sample, independence between the samples, and the Normal/Large Sample condition separately for each group.
Do. The trees were randomly selected from each preserve, supporting generalization to the tree population in each preserve. For the first sample, \(42\leq0.10(420)\), and the population is larger than 420, so the 10% condition is met. For the second, \(36\leq0.10(360)\), and its population is larger than 360, so that sample also meets the condition. Each tree is a separate observational unit, and no tree is measured in both preserves. The two samples come from separate preserves, so the groups can reasonably be treated as independent. Finally, \(42\geq30\) and \(36\geq30\), so the large-sample route is met for each group.
Conclude. The conditions are reasonably supported for a two-sample t procedure comparing mean annual tree growth in the two preserves. The random samples support generalizing to the respective preserve populations; the 10% checks and separate sampling support independence; and both sample sizes meet the large-sample route. This checks whether the procedure is appropriate—it does not itself establish a difference in the means.
Worked Example: Small Samples Require Separate Shape Checks
A school greenhouse manager randomly selects 12 seedlings grown with one watering schedule and 11 seedlings grown with a second schedule. Each seedling is measured once for stem length, in centimeters. The seedlings were selected from separate, large trays, with more than 120 seedlings in the first tray and more than 110 in the second. The seedlings are not matched. Dotplots of both groups show one main cluster, roughly symmetric shapes, and no pronounced outliers.
State. Let \(\mu_1\) and \(\mu_2\) be the population mean stem lengths, in centimeters, for seedlings grown under the first and second schedules. We are checking the conditions for a two-sample t procedure comparing \(\mu_1-\mu_2\).
Plan. Since both sample sizes are less than 30, inspect the shape of each group separately. Also check the selection method, within-group independence, the 10% condition for each sample, and independence between the groups.
Do. Both groups were randomly selected, supporting generalization to seedlings in their respective trays. The first sample satisfies the 10% condition because \(12\leq0.10(120)\), and the actual tray is larger than 120. The second satisfies it because \(11\leq0.10(110)\), and its tray is larger than 110. Each seedling contributes one measurement, and the groups were selected from separate trays without matching, so independence within and between groups is reasonable. Neither sample meets the large-sample route: \(12<30\) and \(11<30\). The stated dotplots therefore matter. Each shows a roughly symmetric, single-cluster pattern without pronounced outliers, which is reasonably consistent with an approximately Normal distribution for that group.
Conclude. The conditions are reasonably supported for comparing the two population mean stem lengths. Because both samples are small, the shape evidence must be reassuring for both groups; here, each group’s own dotplot provides that evidence. A favorable plot for just one group would not be enough.
Worked Example: One Group Has a Concerning Outlier
A fictional recreation program compares the time, in minutes, that two groups of 10 randomly selected participants take to complete a short course. Participants in the groups are different people, selected from separate large rosters. The sample plots show fairly balanced patterns in the first group. In the second group, nine times are close together but one is much larger than the rest, creating a pronounced outlier.
State. Let \(\mu_1\) and \(\mu_2\) be the population mean completion times, in minutes, for the two participant groups. We are assessing whether a two-sample t procedure for \(\mu_1-\mu_2\) is appropriate.
Plan. Check randomness, independence within each group, independence between groups, and Normality or a large sample in each group. Since both samples have fewer than 30 observations, inspect both sample plots.
Do. The participants were randomly selected from their respective rosters, and the groups contain different people with no matching. The rosters are described as large relative to samples of 10, so the 10% condition is met for each sample. These features support randomization, within-group independence, and independence between groups. However, \(10<30\) for each group, so neither group meets the large-sample route. The first group’s plot does not raise a strong shape concern, but the second group’s plot shows a pronounced outlier. The small sample in that group does not provide a large-sample basis for t inference, and its outlier weakens the Normality justification.
Conclude. The random and independence conditions are reasonably supported, but the Normal/Large Sample condition is not supported for the second group. Therefore, a two-sample t procedure is not justified by the information given. It is not enough that the first group’s plot looks suitable; both groups must support the procedure.
Common Mistakes and AP Exam Tips
- Checking Normality only after combining the data. Assess each group separately. State the sample size or describe the graph for each one.
- Using the combined sample size. A total of 60 observations does not meet the large-sample route for two groups of 30 by itself; check each group’s \(n\). If one group has \(n=18\), it remains a small sample.
- Checking the 10% condition only once. For two samples drawn without replacement, compare each sample size with 10% of its own population.
- Assuming equal group sizes make samples independent. Independence comes from the design. The same people measured twice, or deliberately matched people, produce paired data even if group sizes are equal.
- Claiming random selection and random assignment are interchangeable. Random selection supports generalization to a population; random assignment supports a causal comparison of treatments. Explain the actual design.
- Writing only “conditions are met.” A full-credit response names the condition and gives evidence: how units were selected, what the 10% comparison shows, why groups are independent, and what each group’s shape evidence indicates.
A strong condition statement might say: “Both samples were randomly selected from their respective populations. Each sample is no more than 10% of its population, so the 10% condition supports independence within each sample. The samples contain different, unmatched individuals, so the groups can be treated as independent. Because both sample sizes are below 30, I checked each group’s dotplot; both are roughly symmetric with no pronounced outliers, supporting the Normal/Large Sample condition.” Adapt the wording to the evidence. If one group has a concerning feature, report it rather than declaring all conditions satisfied.
Check Your Understanding
For each situation, identify the relevant condition check and explain your reasoning.
- Two independent random samples have sizes 18 and 35. Which group or groups meet the large-sample route?
- Two samples are drawn without replacement from different populations. What population information is needed to check the 10% condition?
- The same 14 people provide one measurement under each of two conditions. Should the groups be treated as independent for a two-sample t procedure? Why or why not?
- One group’s plot is roughly symmetric with no outliers, but the other group has a pronounced outlier and \(n=9\). What does this imply for the Normal/Large Sample condition?
- A randomized experiment assigns different participants to two treatments. What does random assignment support, and does it automatically support generalizing to all people?