Putting the Condition Check Together
In “Writing Condition Checks in Full Sentences,” you practiced connecting each condition to evidence from a study. Here, the task is to assemble those checks efficiently from an exam-style description, including any graphs supplied. The goal is not to list every fact in the prompt. It is to identify the intended procedure, check the relevant evidence in the right order, and reach a careful overall judgment.
A graph matters only after you identify what observations the procedure uses. For a one-sample t procedure, examine the individual sample values. For a paired t procedure, first identify one difference for each pair, then examine the distribution of those differences. The before-and-after measurements are not two independent samples. This distinction, explained in “Conditions for One-Sample Versus Paired Data,” determines which graph is relevant.
This routine is a way to organize the ideas from earlier tutorials, not a reason to rush past evidence. In particular, a graph does not establish random selection or independence, and \(n\geq30\) does not establish that individual observations are Normal. Keep each piece of evidence attached to the condition it can actually address.
A Fast Evidence Map
Under exam conditions, make a brief mental or written map before composing your response. The map has three questions: What are the data? How were the units obtained? What does the sample size or graph show? Answering these in order reduces common mix-ups between study design, independence, and shape.
For a one-sample problem, the observations are the sampled values. For a paired problem, define a consistent difference, such as after minus before, and use the differences.
State whether the study used random sampling or random assignment. Then assess independence among the observations used in the procedure and check the 10% condition if sampling without replacement from a finite population.
If \(n\geq30\), state that the large-sample route supports an approximately Normal sampling distribution of the sample mean. If \(n<30\), describe the relevant graph, including its overall shape and any pronounced outliers.
Say which conditions are supported, unsupported, or impossible to verify from the description. Do not turn a partial check into an unqualified claim that all conditions are met.
For small samples, use the graph to report visible evidence rather than a vague verdict. A dotplot or histogram can show the overall pattern, clusters, gaps, skewness, and observations that stand apart. A Normal probability plot can show whether points follow a roughly straight pattern or depart noticeably from it. As discussed in “Reading a Normal Probability Plot” and “Identifying Skewness and Outliers in Small Samples,” these displays provide evidence about shape; they do not prove the population is Normal.
Do not demand that every sample graph look perfectly Normal. Describe what is present, then say what it suggests for the t procedure. The “Robustness of t Procedures” tutorial explains why modest departures may be less concerning than strong skewness or a pronounced outlier, especially for a small sample. Your condition check should still name the actual evidence rather than rely on the word “robust” as a substitute for examining it.
Worked Examples
Worked Example: A Small One-Sample Check With Two Graphs
A fictional coastal lab randomly selects 16 water-monitoring stations from the 400 stations it operates and measures nitrate concentration, in milligrams per liter, at each station. The selection is without replacement. A dotplot has one main cluster and is roughly symmetric, with no apparent outliers. The Normal probability plot is approximately linear, with small deviations. The lab wants to estimate the mean nitrate concentration at its stations.
State. Let \(\mu\) be the mean nitrate concentration, in milligrams per liter, at the lab’s 400 stations. The proposed procedure is a one-sample t interval based on the 16 station measurements.
Plan. Check random selection, independence using the 10% condition, and the Normal/Large Sample condition. Because \(n=16<30\), assess shape using the graphs of the 16 individual measurements.
Do. The stations were randomly selected, so the Random Condition is supported for generalizing to the lab’s stations. For independence, the observations are from distinct stations, and the 10% condition is met: \(0.10(400)=40\), and \(16\leq40\). Thus the sample is no more than 10% of the station population. Since \(16<30\), the large-sample route does not apply. The dotplot shows one roughly symmetric cluster with no apparent outliers, and the Normal probability plot is approximately linear with only small deviations. Together, these graphs provide reasonable support for the shape condition for a t procedure, although they do not prove the population distribution is Normal.
Conclude. Random selection and the 10% comparison support the random and independence requirements. The small-sample graphs provide reasonable support for the Normal/Large Sample condition. Overall, the evidence supports using a one-sample t interval to estimate the mean nitrate concentration at the lab’s stations.
Worked Example: Checking the Differences in a Paired Study
A fictional school randomly selects 22 students from its 520 students to take part in a study of a new reading routine. Each student completes a timed reading task before and after four weeks of using the routine. The school defines each difference as after minus before, in words read per minute. A dotplot of the 22 differences is roughly mound-shaped, with no apparent outliers, and the Normal probability plot is close to linear. The school wants to estimate the mean change for students at the school.
State. Let \(\mu_d\) be the mean change, in words read per minute, for students at the school. The data are paired, so the proposed procedure is a one-sample t procedure for the 22 student-level differences, calculated as after minus before.
Plan. Check random selection, independence among the students’ differences, and the Normal/Large Sample condition for those differences. If students were sampled without replacement, use the school population size for the 10% check. Do not assess shape by treating all before and after scores as separate observations.
Do. The 22 students were randomly selected, supporting the Random Condition for generalizing to the school’s students. The differences come from 22 distinct students, so independence between differences is plausible if one student’s result does not determine another’s. The sample was drawn without replacement from 520 students, and \(0.10(520)=52\); because \(22\leq52\), the 10% condition is met. Since \(22<30\), the large-sample route does not apply. The dotplot of the differences is roughly mound-shaped with no apparent outliers, and the Normal probability plot is close to linear. These features provide reasonable support for the shape condition for a t procedure.
Conclude. The random selection, plausible independence between students’ differences, and satisfied 10% condition support the design requirements. The graphs of the differences provide reasonable shape evidence for the paired t procedure. The evidence therefore supports using a one-sample t procedure on the differences to estimate the mean change for students at this school. The before-and-after design alone does not establish that the routine caused any observed change.
Worked Example: When a Small-Sample Graph Raises Concern
A fictional community garden records the weekly harvest mass, in kilograms, for 14 plots chosen because they are close to a water source and easy to reach. The garden hopes to estimate the mean weekly harvest mass for all its plots. A histogram of the 14 values is strongly right-skewed, and one value lies far above the rest. The description does not say that plots were randomly selected or give the total number of plots.
State. Let \(\mu\) be the mean weekly harvest mass, in kilograms, for all plots in the garden. The proposed procedure would be a one-sample t procedure using the 14 plot measurements.
Plan. Check whether the selection method supports the Random Condition, whether independence and the 10% condition can be assessed, and whether the shape evidence supports a t procedure. Since \(n=14<30\), the sample graph is especially relevant to the shape check.
Do. The plots were selected for convenience rather than by a stated chance process, so the Random Condition is not supported for generalizing from these plots to all plots in the garden. The plots are distinct, but the description does not provide enough information about the sampling process or plot population to verify independence or make a 10% comparison. Since \(14<30\), the large-sample route does not apply. The histogram is strongly right-skewed and has one value far above the others, providing evidence against the shape condition for a t procedure with this small sample.
Conclude. The selection method does not support generalizing to all garden plots, and the small-sample graph raises a serious shape concern. Independence also cannot be fully verified from the description. A calculator could still produce a t interval, but these conditions do not support treating it as a reliable inference for the mean weekly harvest of all plots.
Common Mistakes and AP Exam Tips
- Checking the original scores instead of the differences. In a paired problem, define the difference clearly and use its graph for the shape check. A graph of the separate before and after values does not answer whether the differences have a suitable shape.
- Letting a graph stand in for the study design. A roughly symmetric dotplot cannot make a convenience sample random. Describe the actual selection method when checking the Random Condition.
- Using the sample size to claim that individual values are Normal. If \(n\geq30\), explain that the large-sample route supports an approximately Normal sampling distribution of the sample mean. If \(n<30\), report what the relevant graph shows.
- Calling a graph “Normal” without describing it. For a small sample, state whether the display is roughly symmetric or mound-shaped, whether the Normal probability plot is approximately linear, and whether there is strong skewness or a pronounced outlier. Use only the features actually shown.
- Writing “the conditions are met” after finding one limitation. Separate the checks. For example, randomness may be supported while independence cannot be verified, or the design may be suitable while the graph raises a shape concern.
- Forgetting units and population. Name the population mean and its units in the opening statement. A condition check should make clear what the proposed inference is about.
A full-credit response is specific without being long-winded. It identifies the procedure and its observations, cites the study’s actual randomization or sampling method, shows a 10% comparison when relevant, and gives concrete graph evidence when the sample is small. It then states what the combined evidence supports and notes any important limitation. The evidence map helps you spend exam time on the condition checks that matter instead of repeating the same sentence for every fact in the prompt.
Check Your Understanding
For each situation, identify the correct observations and state the key condition evidence or limitation.
- A random sample of 18 devices is selected without replacement from 300 devices. A dotplot of battery life is roughly symmetric with no apparent outliers. Which observations should be graphed, and what does the 10% comparison show?
- In a paired study, 21 runners are randomly selected from 240 club members. The graph of their before-minus-after times is strongly left-skewed with one pronounced outlier. Which graph is relevant, and what does it suggest about shape?
- A sample of 35 randomly selected garden beds is used to estimate mean soil acidity. State what \(n\geq30\) supports, and what it does not establish about individual measurements.
- A small sample has a roughly symmetric histogram, but the study used participants who volunteered after seeing a notice. Which condition is not supported for making a population claim, and why can the graph not repair it?
- A study description gives \(n=12\) and a Normal probability plot but no sampling method or population size. Identify one condition the graph can inform and two design checks that the graph cannot complete.