How do you estimate a difference between independent averages?
You will be able to: Construct a two-sample t interval with separate sample variances.
How do you estimate a difference between independent averages?
Two independently sampled groups have mean task times 30 and 26 minutes, with SDs 8 and 6, and 25 observations in each. We want μ₁−μ₂.
A useful starting point: When is a normal model valid for two averages? →
Words and symbols before equations
- Two-sample t interval
- A range for the difference between independent population means.
- Unpooled SE
- √(s₁²/n₁+s₂²/n₂), allowing different population spreads.
- Welch df
- Technology-computed degrees of freedom for this unpooled estimate.
- Difference order
- Group 1 minus group 2, retained throughout.
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. Welch df is used with separate variance estimates; equal population variances are not assumed.
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.
- 95% interval (-0.0294, 8.0294) minutes; margin 4.0294 minutes. 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
Define both populations and the common response. Assume independent random samples from large populations, with small-sample distributions free of strong skewness and outliers.
The point estimate is 30−26=4. SE=√(64/25+36/25)=2 minutes. Technology gives Welch df≈44.51; a 95% critical value is about 2.015.
The interval is approximately 4±4.03, or (−.03,8.03) minutes. Zero is barely inside: these data do not support a two-sided difference at α=.05 under the same procedure. Use unrounded values for decisions.
A worked example, step by step
For the stated summary statistics, lay out the interval calculation.
- Choose a two-sample t interval for μ₁−μ₂ and check both groups.
- Estimate=4 minutes and SE=2 minutes.
- Use Welch df≈44.51 and t*≈2.015.
- The 95% interval is about (−.03,8.03) minutes, including zero.
Do not average the two SDs or assume equal variances without justification.
Does equal sample size imply equal population variance?
Compare with an explanation
No.
Predict. Change one thing. Explain.
Change one group’s SD or size while keeping means fixed. Explain how its separate variance contribution affects interval width.
On narrow screens, swipe or scroll diagrams sideways to read all labels.
95% interval (-0.0294, 8.0294) minutes; margin 4.0294 minutes. 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 |
| t* | 2.01472 |
| 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. Welch df is used with separate variance estimates; equal population variances are not assumed.
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 questionTwo independent suitable samples give mean difference 5, SE=2 and t*=2.05. Construct and interpret the interval.
This response is not submitted or saved. Copy it before leaving.
Compare with the answer and four-point rubric
- 1 point: Use μ₁−μ₂ in the specified order.
- 1 point: ME=2.05×2=4.10.
- 1 point: The interval is (0.90,9.10).
- 1 point: It supports a positive population mean difference at the selected confidence, with scope limited by 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 1Which variances enter the SE?
Both separate sample variances divided by their sample sizes.
RECALL 2What does df determine?
The t critical value or tail probability.
RECALL 3What is the interval’s target?
A difference of population means.
Revisit these tomorrow and a week later. Try a fresh problem and explain why the method applies.
How do you estimate a difference between independent averages?
- Estimate=x̄₁−x̄₂.
- SE=√(s₁²/n₁+s₂²/n₂).
- Interval=(x̄₁−x̄₂)±t*SE.
Remember: Do not average the two SDs or assume equal variances without justification.
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. Welch df is used with separate variance estimates; equal population variances are not assumed.
Refresh Kid · AP Statistics Unit 4 · Objectives 4.7.A, 4.7.B, 4.7.C, 4.7.D · Review edition
Framework, scope and review status
Mapped to College Board, AP Statistics CED, Topic 4.7, objectives 4.7.A, 4.7.B, 4.7.C, 4.7.D. 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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