How do you finish a comparison of two averages?
You will be able to: Calculate a Welch t test and give a qualified contextual conclusion.
How do you finish a comparison of two averages?
Two independent random samples have mean times 30 and 26 minutes, SDs 8 and 6, and n=25 in each. The prespecified question is whether population mean times differ.
A useful starting point: Which checks belong to each independent group? →
Words and symbols before equations
- Welch t statistic
- The observed mean difference divided by the unpooled SE under equality.
- Welch df
- Approximate degrees of freedom calculated from sizes and variance contributions.
- Two-sided p-value
- Null probability of an absolute T at least as large as the observed magnitude.
- Contextual conclusion
- An evidence statement naming both populations and their mean response.
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
Assume large source populations and suitable sample shapes. H₀:μ₁−μ₂=0; Hₐ:μ₁−μ₂≠0. SE=√(64/25+36/25)=2 minutes.
The statistic is t=4/2=2 with Welch df≈44.51. The two-sided p-value is approximately .0516. Use the full technology output rather than rounding a borderline p-value to .05.
At α=.05 fail to reject equality. There is not convincing evidence of different population mean times. This does not prove equality, and it agrees with the 95% interval that just includes zero.
A worked example, step by step
Keep the data fixed but suppose the original prespecified question was whether μ₁>μ₂.
- The one-sided alternative selects the upper tail.
- The statistic remains t=2 with df≈44.51.
- The upper-tail p is approximately .0258.
- At α=.05 reject for this prespecified question; choosing the tail after seeing results would be improper.
Avoid a decision based on a p-value rounded to the significance threshold. Keep sufficient digits.
Does failing to reject show identical means?
Compare with an explanation
No. It means insufficient evidence for the specified alternative.
Predict. Change one thing. Explain.
Keep means fixed, then increase one sample SD. Explain how SE, |t| and p change. State the direction of the evidence in words.
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 questionIndependent suitable samples yield difference 6, SE=2 and a two-sided p=.006. Complete the test conclusion at α=.01.
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Compare with the answer and four-point rubric
- 1 point: State equality null and two-sided alternative for the defined population means.
- 1 point: t=6/2=3; use the appropriate Welch df for the p calculation.
- 1 point: .006≤.01, so reject H₀.
- 1 point: There is evidence the population means differ; describe the positive observed direction and limit causal claims to the design.
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 1What goes in the denominator?
Unpooled SE for the difference.
RECALL 2What does p assume?
The null and model conditions.
RECALL 3What does a nonsignificant test prove?
It does not prove equality.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How do you finish a comparison of two averages?
- t=[(x̄₁−x̄₂)−0]/√(s₁²/n₁+s₂²/n₂).
- Use Welch df and the chosen tail.
- Compare p with α and conclude about population means.
Remember: Avoid a decision based on a p-value rounded to the significance threshold. Keep sufficient digits.
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 4.10.A, 4.10.B, 4.10.C · Review edition
Framework, scope and review status
Mapped to College Board, AP Statistics CED, Topic 4.10, objectives 4.10.A, 4.10.B, 4.10.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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