Chain three sandboxes into one statistical analysis report — draw a sample and read how precise its mean is, turn that precision into a 95% confidence interval, then fit a regression model and read how much of the variance it explains and its effect size. Predict each number before the screen shows it, and record your report at the end.
A real analysis is a report with three stages, and each one hands its number to the next. First you sample: the standard error SE = σ/√n says how precise the sample mean is, and it falls only like 1/√n. Then you infer: that same SE, times a multiplier z*, is the margin of a confidence interval — the honest range for the true mean. Finally you model: an ordinary-least-squares fit is judged by R² = 1 - SSE/SST, the fraction of variance it explains, and by its slope b1, the effect size. This project is the whole arc: the precision you can afford, the interval it buys, and the model you fit — one number feeding the next.
Open each of the three sandboxes in a tab. Each step below names the one sandbox to read and the single choice to make in it; leave every other knob at its default so your numbers match these. The sample runs on the Exponential parent; inference uses the one-sample z test; the model runs on the “Clean linear” preset.
1 CLT sandbox -> Exponential(rate 1), Mean, n = 4
2 Power & CI -> 1-sample z, 2-tailed, sigma = 1, n = 25, alpha = 0.05 3 Regression -> Clean linear, seed 3, OLS
Read four numbers across the three tabs: the standard error with its σ/√n note in the CLT sandbox, the critical value and SE line in the power sandbox (whose product is the CI margin), and the R-squared and fitted slope in the regression workshop.
In the CLT sandbox keep the Exponential(rate 1) parent and the Mean statistic, and set n = 4. That parent has σ = 1, so the standard error of the sample mean is SE = σ/√n = 1/√4. Predict it, then read the spread readout and its σ/√n note.
The σ/√n note reads 0.5. Averaging just four draws already halves the spread of the mean against the parent's — the precision one sample of this size buys you. This is the demo's own SE arithmetic, and it is the number the next stage turns into an interval.
Switch to the power & CI sandbox. Keep the one-sample z test, two-tailed, α = 0.05, with σ = 1 and n = 25. The 95% multiplier is z* = 1.960 and SE = 1/√25 = 0.20, so the margin is m = z*·SE. Predict it, then read the critical-value and SE lines and multiply.
The margin is 0.392, so the 95% interval is x̄ ± 0.392. The precision from stage one (SE = 0.20 here, at n = 25) becomes a concrete range for the true mean. This is the demo's own critical value times its own SE, not a typed number.
Switch to the regression workshop. Load the “Clean linear” preset with seed 3 and fit ordinary least squares. The coefficient of determination R² = 1 - SSE/SST is the fraction of the response's variance the line explains. Predict whether it is close to 1, then read R².
The fit explains R² = 0.9429 of the variance — about 94%, a tight linear relationship. Where stage one measured the spread of a single mean, the model measures how tightly the response tracks its predictor. This is the demo's own analyseFit() output, not a typed number.
Same “Clean linear” preset, seed 3. Beyond how well the line fits, its slope b1 is the effect size — how much y moves per unit of x. Predict roughly the slope of the fitted line, then read b1.
The ordinary-least-squares slope is b1 = 1.2508 — each unit of x lifts y by about 1.25. Slope and R² are different facts: the slope is the effect, the R² is how tightly the points hug it. Together they are the model half of your report. This is the demo's own analyseFit() arithmetic.
Open all three sandboxes and run the report end to end yourself: grow n and watch the standard error fall, feed that SE into the confidence-interval margin and watch it narrow, then fit the model and read the variance it explains and its slope. Each stage hands its number to the next.
A statistical report is a set of choices and the numbers they produce at each stage. Write yours down — fill in each blank from the sandboxes, then add one sentence of reasoning. This is the deliverable; there is no number to read off the screen here.
Stage 1 Sample .... Exponential, n = 4 SE = sigma/sqrt(n) . ______
Stage 2 Infer ..... 95% z, n = 25, sigma = 1 CI margin z*.SE .... ______
Stage 3 Model ..... Clean linear, seed 3, OLS R-squared ......... ______
slope b1 .......... ______
Report note ........ how does the interval width compare with the slope? _____
Design note ........ which stage would you tighten first, and why? __________
Then change one thing — a bigger sample, a different confidence level, a noisier regression preset — and note which of the four numbers moved and by how much. That coupling across sample, infer and model is the whole lesson of the project.