Which checks belong to each independent group?
You will be able to: Verify two-sample t conditions and scope of inference.
Which checks belong to each independent group?
Two schools contribute random samples of 20 and 35 students. Before testing mean study time, the smaller group needs a shape check and each group needs its own sampling-fraction check.
A useful starting point: How do you set up a comparison of independent groups? →
Words and symbols before equations
- Independent groups
- Units belong to separate groups without matched dependence.
- Group-specific shape
- Skewness and outliers are assessed within each sample.
- Random assignment
- Chance allocates treatments; it supports causal comparison.
- Random sampling
- Chance selects population members; it supports generalization.
What this picture assumes
Synthetic summaries, with positive sample SDs. Random samples without replacement use source populations of N=100000 each, satisfying 10% for these sizes. A randomized experiment does not require that sampling-fraction check. Design and shape are assumptions chosen here, not conclusions that summary statistics can verify. Strong skewness/outlier selection conservatively withholds inference at all sizes; real data require examination. Groups are independent. Equality null μ₁−μ₂=0. Welch df uses separate variance estimates, not a pooled SD.
Read the picture in three steps
- Read the axes and labels first. Identify what each symbol and line represents. Read the units and fixed conditions before comparing quantities.
- t=2, df=44.5104, p=0.051625. At α=0.05, fail to reject H₀. Chosen design, sampling fraction where applicable, and shape/size assumptions support the procedure.
- Check what the picture assumes below. Use the Explore task to predict one change before moving a control.
Connect the picture to the mathematics
Use two independent random samples or a properly randomized experiment. Students cannot be quietly reused across groups as if independent.
For samples without replacement, check each n≤.10N. For shape, normal populations support t procedures; with nonnormal populations both sizes should be adequately large. In the small-sample case check both samples for strong skewness and outliers.
The framework’s n≥30 guideline is not permission to ignore extreme data. A randomized experiment without random population sampling can support causation for its experimental setting but may not support broad population generalization.
A worked example, step by step
n₁=20 from N₁=1000 and n₂=35 from N₂=2000 are independent SRSs; both sample distributions have no strong skewness or outliers.
- Independent random sampling is given.
- 20≤100 and 35≤200 pass the separate 10% checks.
- The stated shapes support the small-sample t approximation.
- The inference can target the sampled populations, but observational sampling alone does not establish causation.
A total of 55 observations cannot replace checking each group and its collection process.
Does a randomized trial automatically represent every student in the US?
Compare with an explanation
No. Recruitment and target population determine generalizability.
Predict. Change one thing. Explain.
Switch to an unsuitable collection or small, skewed sample assumption. Explain why a numerically computable t is not enough for justified inference.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
t=2, df=44.5104, p=0.051625. At α=0.05, fail to reject H₀. Chosen design, sampling fraction where applicable, and shape/size assumptions support the procedure.
| Quantity | Value |
|---|---|
| Estimate (minutes) | 4 |
| SE (minutes) | 2 |
| Degrees of freedom | 44.51039 |
| Group 1 variance contribution (min²) | 2.56 |
| Group 2 variance contribution (min²) | 1.44 |
Synthetic summaries, with positive sample SDs. Random samples without replacement use source populations of N=100000 each, satisfying 10% for these sizes. A randomized experiment does not require that sampling-fraction check. Design and shape are assumptions chosen here, not conclusions that summary statistics can verify. Strong skewness/outlier selection conservatively withholds inference at all sizes; real data require examination. Groups are independent. Equality null μ₁−μ₂=0. Welch df uses separate variance estimates, not a pooled SD.
Explain what you noticed: Answer the investigation prompt above. State one observation and explain it using the means, standard errors, pairing, graph scales or model assumptions. Identify what the representation cannot tell you.
Apply the idea to a fresh problem Practice →Show what you understand.
Two original questions are a starting check, not proof of mastery. Explain your choice before revealing the answer.
Original written challenge
4 points · self-check · not an official AP questionA randomized experiment assigns 40 volunteers per group. Explain the randomization, size and scope issues.
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Compare with the answer and four-point rubric
- 1 point: Random assignment supports treatment comparison.
- 1 point: Each group has 40 units, supporting a usual size guideline while retaining outlier checks.
- 1 point: A 10% population-sampling condition is not required merely for treatment assignment.
- 1 point: Volunteers may limit generalization beyond the experimental setting.
Accept equivalent correct methods and explanations. This is a Refresh Kid teaching rubric, not an official AP scoring guideline.
Retrieve it before you reveal it.
RECALL 1Which checks are repeated for both samples?
Sampling fraction and shape/size.
RECALL 2Does assignment equal sampling?
No; they support different conclusions.
RECALL 3Should extreme data be ignored above n=30?
No.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
Which checks belong to each independent group?
- Verify randomization and independence.
- Sampling without replacement: each n≤.10N.
- Check both distributions, especially if either n<30.
Remember: A total of 55 observations cannot replace checking each group and its collection process.
Conditions: Synthetic summaries, with positive sample SDs. Random samples without replacement use source populations of N=100000 each, satisfying 10% for these sizes. A randomized experiment does not require that sampling-fraction check. Design and shape are assumptions chosen here, not conclusions that summary statistics can verify. Strong skewness/outlier selection conservatively withholds inference at all sizes; real data require examination. Groups are independent. Equality null μ₁−μ₂=0. Welch df uses separate variance estimates, not a pooled SD.
Refresh Kid · AP Statistics Unit 4 · Objectives 4.9.C · Review edition
Framework, scope and review status
Mapped to College Board, AP Statistics CED, Topic 4.9, objectives 4.9.C. Framework effective Fall 2026, checked September 17, 2026. Unit 4 includes sampling distributions of means, one-sample and paired t inference, and independent two-sample t inference; it is part of the revised five-unit course.
Examples and datasets are synthetic, independently authored teaching material. Mean inference requires a justified design and suitable shape or sample size. Paired analysis uses one sample of differences. Independent two-sample inference uses separate variance estimates and technology-computed Welch degrees of freedom. Extreme skewness and influential observations need attention even in larger samples. This model conservatively withholds inference when those warnings are selected. Conclusions are limited by random sampling and/or assignment as appropriate.
The Organic Chemistry Tutor companion title and destination were located; the full video was not reviewed. Khan Academy’s destination was checked, but its lesson content was not fully readable by the research tool. OpenStax provides optional reference reading. No provider scripts, questions or graphics were copied. Refresh Kid is not affiliated with these providers.
GitHub’s 3D website collection informed optional spatial inspection. Our original paired-data display uses self-hosted Three.js with its MIT license. Two measurement columns are connected within each labeled student lane; horizontal position is before/after, vertical position is time, and depth separates identities rather than representing a numerical variable. Rotation can separate overlapping connectors. Exact values and differences always remain in the 2D table and labeled plot. The broken-matching option is an explicit counterexample, not a legitimate alternative analysis. No autoplay; complete teaching remains available without 3D.
Independent teacher review and observation of students remain pending. Technical checks do not certify statistical accuracy, accessibility or learning effectiveness. This is a review edition.
Released AP Statistics questions and scoring guides are optional. Older exams use the earlier framework, so check alignment before selecting parts. All practice on this page is original, not official AP material.
Learn → Explore → Practice → Review is informed by the IES learning guide. This implementation has not yet been evaluated with learners.
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