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OPEN MATHEMATICS · STATISTICS
Center and spread
Student: _______________________________________ Date: __________________
1. Predict before touching the controls
For 40, 45, 50, 55, and 60, what are the mean and median? What if the final value becomes 600?
2. Run the experiment
Move only the fifth observation and read the mean, median, and sample standard deviation.
3. Explain the result
Explain which summary stays resistant and why neither a mean nor median alone tells the whole distribution.
4. One important limitation
Which assumption, sampling decision, or source of uncertainty would matter if you used this result to make a real-world claim?
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OPEN MATHEMATICS · STATISTICS
Visualizing distributions
Student: _______________________________________ Date: __________________
1. Predict before touching the controls
Will moving from four to sixteen histogram bins change the original observations, mean, or quartiles?
2. Run the experiment
Hold the synthetic data fixed while adjusting the number of equal-width bins; expand the frequency table.
3. Explain the result
Describe how a visualization choice changes perceived structure without changing the raw data.
4. One important limitation
Which assumption, sampling decision, or source of uncertainty would matter if you used this result to make a real-world claim?
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OPEN MATHEMATICS · STATISTICS
Correlation and outliers
Student: _______________________________________ Date: __________________
1. Predict before touching the controls
Can moving one observation alter the sample Pearson correlation? Can it establish a cause?
2. Run the experiment
Move observation eight above and below the seven-point trend; inspect r and the full table.
3. Explain the result
Describe influence on a linear summary and explain why observational association is not a causal effect.
4. One important limitation
Which assumption, sampling decision, or source of uncertainty would matter if you used this result to make a real-world claim?
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OPEN MATHEMATICS · STATISTICS
Survey nonresponse selection
Student: _______________________________________ Date: __________________
1. Predict before touching the controls
A population's views are fixed at 54% support. Could an honest survey report 63% without anyone changing their mind?
2. Run the experiment
Change the two groups' response probabilities; compare the respondent-weighted percentage with the fixed population percentage.
3. Explain the result
Use the expected response counts to explain group-composition bias; distinguish it from random sampling error.
4. One important limitation
Which assumption, sampling decision, or source of uncertainty would matter if you used this result to make a real-world claim?
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OPEN MATHEMATICS · STATISTICS
Central limit theorem
Student: _______________________________________ Date: __________________
1. Predict before touching the controls
If individual observations come from a strongly skewed distribution, must their sample means also be just as skewed?
2. Run the experiment
Choose an exponential population; compare means from samples of 1, 5, 30, and 80 observations.
3. Explain the result
Identify which distribution becomes approximately normal and which remains right-skewed; discuss sample-size assumptions.
4. One important limitation
Which assumption, sampling decision, or source of uncertainty would matter if you used this result to make a real-world claim?
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OPEN MATHEMATICS · STATISTICS
Normal probability
Student: _______________________________________ Date: __________________
1. Predict before touching the controls
What proportion lies between −1.96 and +1.96 under a standard normal model?
2. Run the experiment
Move the symmetric interval endpoints and watch both the curve area and numerical probability.
3. Explain the result
Distinguish density height, interval probability, and the interpretation of a confidence interval.
4. One important limitation
Which assumption, sampling decision, or source of uncertainty would matter if you used this result to make a real-world claim?
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OPEN MATHEMATICS · STATISTICS
Bayesian base rates
Student: _______________________________________ Date: __________________
1. Predict before touching the controls
With 1% prevalence, 90% sensitivity, and 95% specificity, will most positive screening results be genuine?
2. Run the experiment
Hold test properties fixed and increase prevalence; inspect true and false positive expected counts.
3. Explain the result
Explain in ordinary language why sensitivity does not equal positive predictive value. The example is fictional.
4. One important limitation
Which assumption, sampling decision, or source of uncertainty would matter if you used this result to make a real-world claim?
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OPEN MATHEMATICS · STATISTICS
Confidence interval coverage
Student: _______________________________________ Date: __________________
1. Predict before touching the controls
If a 95% confidence procedure makes fifty intervals, must exactly forty-eight contain the true parameter?
2. Run the experiment
Run several batches at 95%, then change to 90% or 99% and compare misses and interval width.
3. Explain the result
Explain the fixed-parameter/repeated-random-interval framework and why observed coverage fluctuates.
4. One important limitation
Which assumption, sampling decision, or source of uncertainty would matter if you used this result to make a real-world claim?
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OPEN MATHEMATICS · STATISTICS
Exact binomial hypothesis test
Student: _______________________________________ Date: __________________
1. Predict before touching the controls
Under a fair-coin null model, should 18 heads in 20 tosses be surprising on a two-sided test?
2. Run the experiment
Change the observed count while preserving the null model, and identify the shaded extreme outcomes.
3. Explain the result
Interpret the exact p-value conditional on the null; distinguish it from P(null given data).
4. One important limitation
Which assumption, sampling decision, or source of uncertainty would matter if you used this result to make a real-world claim?
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OPEN MATHEMATICS · STATISTICS
Simpson's paradox
Student: _______________________________________ Date: __________________
1. Predict before touching the controls
Could one recorded treatment success percentage be higher in both stone-size groups but lower overall?
2. Run the experiment
Switch between pooled and stone-size-specific historical success proportions.
3. Explain the result
Explain the differing case mixes and why neither pooled nor stratified observational rates prove causal superiority.
4. One important limitation
Which assumption, sampling decision, or source of uncertainty would matter if you used this result to make a real-world claim?
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OPEN MATHEMATICS · STATISTICS
Bootstrap sampling
Student: _______________________________________ Date: __________________
1. Predict before touching the controls
If you repeatedly resample the same twelve observed waits with replacement, will every replicate have the same mean?
2. Run the experiment
Compare 200 and 3,000 replicates, inspect interval endpoints and the effect of the 39-minute wait.
3. Explain the result
Separate the bootstrap's Monte Carlo variability from information missing in the original sample.
4. One important limitation
Which assumption, sampling decision, or source of uncertainty would matter if you used this result to make a real-world claim?
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OPEN MATHEMATICS · STATISTICS
Exact randomization inference
Student: _______________________________________ Date: __________________
1. Predict before touching the controls
For eight people randomly assigned in groups of four, how many balanced allocations are possible?
2. Run the experiment
Compare the two-sided and one-sided exact p-values for the fixed fictional outcomes.
3. Explain the result
Explain how design-based inference relies on the actual random assignment mechanism and pre-specified alternative.
4. One important limitation
Which assumption, sampling decision, or source of uncertainty would matter if you used this result to make a real-world claim?
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OPEN MATHEMATICS · STATISTICS
Regression residual diagnostics
Student: _______________________________________ Date: __________________
1. Predict before touching the controls
Can the least-squares algorithm fit the optimal straight line and still miss a curved relationship?
2. Run the experiment
Increase the synthetic data curvature and compare fitted lines, R², and residual structure.
3. Explain the result
Describe a systematic residual pattern and why a high in-sample R² is not proof of causality or model validity.
4. One important limitation
Which assumption, sampling decision, or source of uncertainty would matter if you used this result to make a real-world claim?
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