High-school Statistics and Probability objective evidence

Choose a display that preserves the question — Statistics foundations: data, design and sampling variability

• Using bins [0,5), [5,10), [10,15], how many of 0,4,5,5,9,10,15 lie in the second bin?Automatically checked response; reasoning still matters
• For 1,2,4,7,8,9,10,15, use medians of the lower and upper halves. Find IQR.Automatically checked response; reasoning still matters
• A histogram bin of width 4 contains 12 observations. What frequency-density height gives area equal to count?Automatically checked response; reasoning still matters
• Counts are 20 in [0,2) and 30 in [2,8). Explain why drawing heights 20 and 30 is misleading when comparing areas, and give density heights.Written reasoning: review required

Statistics: design, variation and model selection — Statistics foundations: data, design and sampling variability

• For data 2, 4, 4, 6, 9, 11, compute mean, median and sample variance. Use the median-of-halves convention for quartiles and the 1.5 IQR rule for outliers. Describe what changes if 11 is replaced by 41 and why the original boxplot cannot reveal whether the two 4s were repeats from the same person.Written reasoning: review required

Construct three views of one dataset — Statistical investigations, simulation and probability decisions

• For synthetic waits 2,2,3,4,4,4,6,7,9,15 minutes, construct a dotplot, histogram with bins [0,4),[4,8),[8,12),[12,16), and a modified boxplot. Supply labeled graphs or precise coordinate descriptions and state conventions.Written reasoning: review required
• A histogram uses bins labeled 0–5 and 5–10 and assigns a value 5 to both. Repair the construction for values 1,5,5,7,10 and show counts under an explicit nonoverlapping convention.Written reasoning: review required
• A histogram uses [0,2) with count 6 and [2,8) with count 12. Construct a frequency-density version and explain why raw heights 6 and 12 misrepresent the comparison of concentrations.Written reasoning: review required
• A teammate removes the 15-minute wait from the first dataset because it appears beyond a boxplot fence. Explain the appropriate next step and the limitations of the boxplot.Written reasoning: review required

Produce and defend a statistical investigation — Statistical investigations, simulation and probability decisions

• Synthetic package delays are 0,1,1,2,2,3,4,4,7,12 minutes. Construct a dotplot, histogram with bins [0,3),[3,6),[6,9),[9,12] (last endpoint included), and a modified boxplot using median-of-halves quartiles. Explain an unusual value without automatically deleting it.Written reasoning: review required

Summarize variation without hiding extremes — Statistics foundations: data, design and sampling variability

• Two delivery groups both have mean20 minutes. Group A is 19,20,21 and B is 5,20,35. Compare their consistency and the limits of these data.Written reasoning: review required
• Replace 10 by 100 in 2,4,4,10. Which remains unchanged?Automatically checked response; reasoning still matters
• For 1,3,5, compare population variance when these are the entire population with sample variance when used for estimating a larger population variance.Written reasoning: review required

Compare shapes before choosing summaries — Statistical investigations, simulation and probability decisions

• Compare A:4,5,5,6,6,7,7,8 and B:2,3,3,4,4,5,8,19 minutes with aligned dotplots and boxplots. Choose summaries for a typical wait and variability; explain why a mean-only ranking fails.Written reasoning: review required
• For the preceding A/B datasets, one customer prioritizes a typical short wait and another must avoid waits above 10 minutes. Compare the observed fractions and explain why neither sample proves future reliability.Written reasoning: review required
• Every wait in a dataset is converted from minutes to seconds. Explain the changes to mean, median, IQR, standard deviation and variance; distinguish conversion from adding a fixed delay.Written reasoning: review required

Produce and defend a statistical investigation — Statistical investigations, simulation and probability decisions

• Compare synthetic completion times A:5,6,6,7,7,8,8,9 and B:2,3,4,4,5,6,10,22 minutes using common-scale displays. Choose summaries for typical time and spread, then discuss a deadline of 9 minutes and sampling scope.Written reasoning: review required

Choose a display that preserves the question — Statistics foundations: data, design and sampling variability

• Explain why two groups with identical five-number summaries can have different distribution shapes.Written reasoning: review required

Summarize variation without hiding extremes — Statistics foundations: data, design and sampling variability

• Two delivery groups both have mean20 minutes. Group A is 19,20,21 and B is 5,20,35. Compare their consistency and the limits of these data.Written reasoning: review required

Statistics: design, variation and model selection — Statistics foundations: data, design and sampling variability

• For data 2, 4, 4, 6, 9, 11, compute mean, median and sample variance. Use the median-of-halves convention for quartiles and the 1.5 IQR rule for outliers. Describe what changes if 11 is replaced by 41 and why the original boxplot cannot reveal whether the two 4s were repeats from the same person.Written reasoning: review required

Compare shapes before choosing summaries — Statistical investigations, simulation and probability decisions

• Compare A:4,5,5,6,6,7,7,8 and B:2,3,3,4,4,5,8,19 minutes with aligned dotplots and boxplots. Choose summaries for a typical wait and variability; explain why a mean-only ranking fails.Written reasoning: review required
• For the preceding A/B datasets, one customer prioritizes a typical short wait and another must avoid waits above 10 minutes. Compare the observed fractions and explain why neither sample proves future reliability.Written reasoning: review required
• A report compares morning observations from service A with peak-evening observations from service B and credits staffing for a difference. Design a fairer descriptive comparison and limit any causal claim.Written reasoning: review required

Produce and defend a statistical investigation — Statistical investigations, simulation and probability decisions

• Compare synthetic completion times A:5,6,6,7,7,8,8,9 and B:2,3,4,4,5,6,10,22 minutes using common-scale displays. Choose summaries for typical time and spread, then discuss a deadline of 9 minutes and sampling scope.Written reasoning: review required

Locate a value within a model — Statistics foundations: data, design and sampling variability

• A standard normal table gives P(Z≤1.2)=0.8849. Find P(Z>1.2).Automatically checked response; reasoning still matters
• A waiting-time dataset has many near-zero values and a long right tail. Why can you not use the 68–95–99.7 rule merely because its mean and SD are known?Written reasoning: review required
• A student scores80 where mean70 and SD5; another scores90 where mean80 and SD20. Compare standardized positions without claiming equal subject proficiency.Written reasoning: review required

Statistics: design, variation and model selection — Statistics foundations: data, design and sampling variability

• A synthetic length model is normal with mean 50 mm and SD 4 mm. Given P(Z≤1.5)=0.9332 and P(Z≤−1)=0.1587, find P(46≤X≤56). Convert to Y=2X+3: state its mean, SD and the corresponding interval. Explain whether these calculations validate the normal model.Written reasoning: review required

Check a normal model against raw observations — Statistical investigations, simulation and probability decisions

• Enter 40,45,45,50,50,50,55,55,60 in technology. Record n, mean, sample SD and a dotplot/histogram. Assess a normal approximation and estimate the proportion of a comparable population above 56.1237 using Φ(1)=0.8413.Written reasoning: review required
• For synthetic delay values 0,0,0,1,1,2,2,4,30, make a display and evaluate a fitted normal model for predicting delay probabilities. Explain why computing mean and SD successfully does not validate the model.Written reasoning: review required
• A stated normal model has mean 100 and SD 12. Estimate the proportion between 88 and 124 using Φ(−1)=0.1587 and Φ(2)=0.9772. Explain why sample SD would introduce another layer of uncertainty.Written reasoning: review required
• A tool reports SD with divisor n while a report requests sample SD with divisor n−1. For n=10 and squared-deviation sum 90, explain both outputs and repair the report.Written reasoning: review required

Produce and defend a statistical investigation — Statistical investigations, simulation and probability decisions

• Enter synthetic readings 62,66,68,70,70,70,72,74,78 into technology. Record n, mean, sample SD and a display. Assess a normal approximation and estimate the model fraction below 70−√21 using Φ(−1)=0.1587. Contrast the observed fraction below that cutoff and critique certainty.Written reasoning: review required

Change the denominator deliberately — Statistics foundations: data, design and sampling variability

• Construct a two-way table: 40 cyclists include 12 helmet nonusers; 60 walkers include 9 hat nonusers. Explain why “helmet versus hat use” needs a measurement warning.Written reasoning: review required
• Group A has 18 successes and 12 failures. What is the success proportion within A?Automatically checked response; reasoning still matters
• A has 18 successes; B has 42 successes. Among successes, what fraction belongs to A?Automatically checked response; reasoning still matters
• A has 30 members and B has 70. If 18 A members succeed, what fraction of the entire sample is both A and successful?Automatically checked response; reasoning still matters
• A succeeds 18/30 of the time and B succeeds 42/70. In this table:Automatically checked response; reasoning still matters

Statistics: design, variation and model selection — Statistics foundations: data, design and sampling variability

• Among 80 morning commuters, 48 use the bus; among 120 afternoon commuters, 42 use it. Construct the full two-way count table. Find P(bus), P(morning | bus), and P(bus | morning). Are time group and bus use independent in this empirical distribution? Explain why reversing a conditional is not justified.Written reasoning: review required

Describe a relationship before fitting it — Statistics foundations: data, design and sampling variability

• Show why (−1,1),(0,0),(1,1) has zero correlation but a deterministic nonlinear relationship.Written reasoning: review required
• A new point lies far to the right of an existing cloud. Explain why distance in x alone does not determine whether correlation becomes stronger.Written reasoning: review required
• Which is compatible with r=0?Automatically checked response; reasoning still matters

Fit a line and inspect its failures — Statistics foundations: data, design and sampling variability

• Fit a least-squares line to (0,1),(1,2),(2,5), and report the three residuals.Written reasoning: review required
• A line is fitted to battery performance at temperatures10–30°C. Evaluate a prediction at80°C and a causal claim from its slope.Written reasoning: review required

Change units without changing the question accidentally — Statistics foundations: data, design and sampling variability

• A fitted model is log10(y)=1+0.2x for positive y. Interpret a one-unit increase in x and explain why back-transforming fitted logs need not estimate the arithmetic conditional mean.Written reasoning: review required

Compare fitted models and plotted residuals — Statistical investigations, simulation and probability decisions

• Enter synthetic elapsed-day/response pairs (0,3),(1,4),(2,8),(3,12),(4,20),(5,29). Use technology to fit linear, quadratic and log-linear exponential candidates. Record each equation, original-response-scale SSE, scatterplot and residual plot. Recommend a tentative model with limitations.Written reasoning: review required
• A fitted delivery-time model is ŷ=12+0.8x, with x in kilometers and y in minutes, based on trips from 5 to 30 km. Interpret slope/intercept and assess prediction at x=0 and x=100.Written reasoning: review required
• An exponential fit log y=α+βx is attempted on y values −1,0,2,5. Explain why the workflow fails and why adding an arbitrary constant is not a neutral repair.Written reasoning: review required

Produce and defend a statistical investigation — Statistical investigations, simulation and probability decisions

• For synthetic time/response pairs (0,5),(1,6),(2,11),(3,14),(4,23),(5,31), use technology to compare linear, quadratic and exponential fits. Supply original-scale SSEs and residual plots. Interpret a linear slope’s units if time is days and response is millimeters, and discuss the intercept and extrapolation to day 20.Written reasoning: review required

Fit a line and inspect its failures — Statistics foundations: data, design and sampling variability

• The fitted line is ŷ=3+1.5x. At x=6 the observed y is10. Find the residual.Automatically checked response; reasoning still matters
• A residual plot shows a systematic U shape. What is a reasonable response?Automatically checked response; reasoning still matters
• Fit a least-squares line to (0,1),(1,2),(2,5), and report the three residuals.Written reasoning: review required

Statistics: design, variation and model selection — Statistics foundations: data, design and sampling variability

• Four paired observations are (0,1),(1,3),(2,2),(3,6). Compute the least-squares slope and intercept from centered sums, and the residuals. Calculate r² using the centered sums. Explain what residual inspection and the observational design can and cannot tell you about a causal claim or prediction at x=30.Written reasoning: review required

Compare fitted models and plotted residuals — Statistical investigations, simulation and probability decisions

• Enter synthetic elapsed-day/response pairs (0,3),(1,4),(2,8),(3,12),(4,20),(5,29). Use technology to fit linear, quadratic and log-linear exponential candidates. Record each equation, original-response-scale SSE, scatterplot and residual plot. Recommend a tentative model with limitations.Written reasoning: review required
• A linear fit yields ordered residuals at x=0,1,2,3,4 of 2,−1,−2,−1,2. Construct its residual plot and explain what it suggests, what it does not prove, and a useful next comparison.Written reasoning: review required

Produce and defend a statistical investigation — Statistical investigations, simulation and probability decisions

• For synthetic time/response pairs (0,5),(1,6),(2,11),(3,14),(4,23),(5,31), use technology to compare linear, quadratic and exponential fits. Supply original-scale SSEs and residual plots. Interpret a linear slope’s units if time is days and response is millimeters, and discuss the intercept and extrapolation to day 20.Written reasoning: review required

Fit a line and inspect its failures — Statistics foundations: data, design and sampling variability

• Fit a least-squares line to (0,1),(1,2),(2,5), and report the three residuals.Written reasoning: review required
• A line is fitted to battery performance at temperatures10–30°C. Evaluate a prediction at80°C and a causal claim from its slope.Written reasoning: review required

Statistics: design, variation and model selection — Statistics foundations: data, design and sampling variability

• Four paired observations are (0,1),(1,3),(2,2),(3,6). Compute the least-squares slope and intercept from centered sums, and the residuals. Calculate r² using the centered sums. Explain what residual inspection and the observational design can and cannot tell you about a causal claim or prediction at x=30.Written reasoning: review required

Fit a line and inspect its failures — Statistics foundations: data, design and sampling variability

• For three pairs, Σ(x−x̄)²=8 and Σ(x−x̄)(y−ȳ)=12. Find the least-squares slope.Automatically checked response; reasoning still matters
• A least-squares line has slope3/2, x̄=4 and ȳ=9. Find its intercept.Automatically checked response; reasoning still matters
• A line is fitted to battery performance at temperatures10–30°C. Evaluate a prediction at80°C and a causal claim from its slope.Written reasoning: review required

Compare fitted models and plotted residuals — Statistical investigations, simulation and probability decisions

• A fitted delivery-time model is ŷ=12+0.8x, with x in kilometers and y in minutes, based on trips from 5 to 30 km. Interpret slope/intercept and assess prediction at x=0 and x=100.Written reasoning: review required

Produce and defend a statistical investigation — Statistical investigations, simulation and probability decisions

• For synthetic time/response pairs (0,5),(1,6),(2,11),(3,14),(4,23),(5,31), use technology to compare linear, quadratic and exponential fits. Supply original-scale SSEs and residual plots. Interpret a linear slope’s units if time is days and response is millimeters, and discuss the intercept and extrapolation to day 20.Written reasoning: review required

Describe a relationship before fitting it — Statistics foundations: data, design and sampling variability

• For paired points (1,8),(2,6),(3,4),(4,2), find Pearson correlation r.Automatically checked response; reasoning still matters
• A dataset has r=3/5. Convert positive distance measurements from metres to centimetres. What is the new r?Automatically checked response; reasoning still matters
• Show why (−1,1),(0,0),(1,1) has zero correlation but a deterministic nonlinear relationship.Written reasoning: review required

Statistics: design, variation and model selection — Statistics foundations: data, design and sampling variability

• Four paired observations are (0,1),(1,3),(2,2),(3,6). Compute the least-squares slope and intercept from centered sums, and the residuals. Calculate r² using the centered sums. Explain what residual inspection and the observational design can and cannot tell you about a causal claim or prediction at x=30.Written reasoning: review required

Calculate and interpret Pearson correlation — Statistical investigations, simulation and probability decisions

• Enter paired synthetic study-hours x=1,2,3,4,5 and scores y=2,4,5,4,7 in PairedDataWorkbench or equivalent technology. Record the input, a scatterplot, Pearson r and an independent sum check. Interpret the result without a causal claim.Written reasoning: review required
• If hours are converted to minutes and every score is increased by 10, what happens to r? What if scores are multiplied by −1 instead? Explain from centered quantities.Written reasoning: review required
• Enter x=−2,−1,0,1,2 and y=4,1,0,1,4. Construct the scatterplot and explain why Pearson r=0 does not mean no relationship. Compare with a constant-y dataset.Written reasoning: review required
• A spreadsheet sorts x values but not their paired y values before computing correlation. Explain the failure and propose a reproducible repair and a check for an influential observation.Written reasoning: review required

Produce and defend a statistical investigation — Statistical investigations, simulation and probability decisions

• Use paired synthetic inputs x=0,1,2,3,4 and y=1,3,2,6,8. Compute Pearson r with technology and provide a scatterplot and centered-sum check. Interpret association, then explain the effect of converting x to 60x and sorting only one column.Written reasoning: review required

Describe a relationship before fitting it — Statistics foundations: data, design and sampling variability

• Ice cream sales and swimming incidents rise together. Which follows from correlation alone?Automatically checked response; reasoning still matters

Use design to support a causal claim — Statistics foundations: data, design and sampling variability

• Students choose whether to attend an optional revision session. Attenders score higher. Explain the confounding and propose a stronger study.Written reasoning: review required
• Volunteers are randomly assigned to two tutoring methods. The strongest design-based claim is:Automatically checked response; reasoning still matters

Choose who can enter the sample — Statistics foundations: data, design and sampling variability

• A city surveys voters using landline numbers, then reports attitudes of all adult residents. Identify two distinct selection problems and a repair.Written reasoning: review required
• Describe a cluster sample of apartment residents and contrast it with sampling within every building.Written reasoning: review required

Inference: choose methods and defend conclusions — Statistical inference: estimates, tests and defensible conclusions

• A web poll of 10,000 volunteers reports 62% support and a very narrow nominal normal interval. A different study randomly assigns 80 volunteers to two teaching methods and measures exam scores. Write a short decision brief: what can each study support, what must be checked, and what additional design would support generalization to all students?Written reasoning: review required

Build a coherent probability model — Statistics foundations: data, design and sampling variability

• A die shows six on30 of100 rolls. Explain why the observed proportion does not prove either that the die is fair or that P(six)=0.30.Written reasoning: review required

Test a proposed success probability — Statistical inference: estimates, tests and defensible conclusions

• In a representative independent sample of 200, 70 successes occur. Test H0:p=0.4 versus p<0.4. Give z and a conclusion using supplied lower-tail p≈0.0745 for z≈−1.443; α=0.05.Written reasoning: review required
• A machine produces 200 items in one tightly coupled batch. Why may a binomial z test overstate evidence even with large success and failure counts?Written reasoning: review required

Specify a test before seeing its answer — Statistical inference: estimates, tests and defensible conclusions

• A prespecified test yields p=0.12 at α=0.05. Write a conclusion without accepting the null as proven.Written reasoning: review required

Use design to support a causal claim — Statistics foundations: data, design and sampling variability

• Volunteers are randomly assigned to two tutoring methods. The strongest design-based claim is:Automatically checked response; reasoning still matters
• Plan a matched-pairs test of two keyboard layouts on the same participants, addressing order effects.Written reasoning: review required
• Students choose whether to attend an optional revision session. Attenders score higher. Explain the confounding and propose a stronger study.Written reasoning: review required

Statistics: design, variation and model selection — Statistics foundations: data, design and sampling variability

• A school invites volunteers through its sports newsletter to compare two study reminders. It randomly assigns each volunteer one reminder and measures the number of assignments submitted on time. Define the observational unit, variables and a statistical question. Explain which causal and population claims the design could support, name two selection limitations, and propose a design that improves generalization without losing random assignment.Written reasoning: review required

Distinguish the data from the statistic — Statistics foundations: data, design and sampling variability

• Describe three different distributions in a study of heights using repeated random samples of25 adults.Written reasoning: review required
• An estimator always reports the sample mean plus5. Under independent sampling with population mean μ, assess its bias and whether increasing n removes it.Written reasoning: review required

Decide what the evidence can support next — Statistics foundations: data, design and sampling variability

• A random sample has72 successes in120 observations. Find the sample proportion.Automatically checked response; reasoning still matters
• A voluntary poll of10,000 visitors yields a tiny reported standard error. Audit a claim about all city residents.Written reasoning: review required

Interpret an interval as a procedure — Statistical inference: estimates, tests and defensible conclusions

• Repair: “There is a 95% chance that 95% of people have values between our mean-confidence bounds.”Written reasoning: review required

Estimate a population proportion — Statistical inference: estimates, tests and defensible conclusions

• For 50 successes in 100 independent representative trials, calculate the approximate 95% Wald interval using z*=1.96. Check counts and interpret.Written reasoning: review required

Develop a margin from repeated samples — Statistical investigations, simulation and probability decisions

• Use SamplingMargin with n=10, 200 training trials and target 90% coverage. Record seed, error-frequency table, selected nearest-rank margin and at least three trace rows. Evaluate that fixed margin on the separate evaluation seed with 200 trials. Explain why the two coverage fractions need not agree.Written reasoning: review required
• A random survey has estimate 0.55 and an illustrative simulation-calibrated absolute margin 0.20. Give the interval and explain exactly what assumptions are needed to interpret it. Why is simulation from a known p=0.4 not a universal guarantee for every unknown population?Written reasoning: review required
• Calibrate at population p=0.4, then evaluate with selected-subgroup p=0.7 while retaining error relative to population 0.4. Compare n=10 and n=40 using recorded trials. Explain why a larger sample need not repair the problem.Written reasoning: review required
• A training run selects a 95% nearest-rank margin and reports at least 95% training coverage. Explain why this does not independently validate 95% future coverage, and propose checks.Written reasoning: review required

Produce and defend a statistical investigation — Statistical investigations, simulation and probability decisions

• Use SamplingMargin to calibrate a 90% margin at n=20 with 200 training draws, then evaluate it on 200 separate draws. Repeat at n=40. Record seeds, selected ranks/margins, error tables and evaluation coverage. Connect an illustrative observed estimate 0.60 to each interval and distinguish four uncertainty sources.Written reasoning: review required

Build a null distribution from the assignment mechanism — Statistical inference: estimates, tests and defensible conclusions

• Outcomes are 1,2,4,5, with two treated units. Under the sharp null, enumerate treatment-minus-control mean differences for the six allocations. If treated outcomes are 4,5, find the exact upper-tail p-value.Written reasoning: review required
• Why is unrestricted label permutation not automatically justified for two observational groups?Written reasoning: review required

Inference: choose methods and defend conclusions — Statistical inference: estimates, tests and defensible conclusions

• Four fixed outcomes are 1,3,6,8. Exactly two units were randomly assigned treatment. Under the sharp null of no effect for any unit, enumerate treatment-minus-control mean differences. The observed treated units have outcomes 6 and 8. Find the exact two-sided randomization p-value using absolute differences and including ties. Explain why observationally chosen groups would not justify the same reference.Written reasoning: review required

Simulate treatment reassignment under a null — Statistical investigations, simulation and probability decisions

• Run RandomizationExplorer with its six fixed outcomes and three units per group. Before drawing, record the observed assignment and two-sided |difference| tail rule. Run 200 seeded draws, retain three assignment/statistic rows and the tail fraction, then compare with the exact 20-assignment reference.Written reasoning: review required
• For distinct outcomes 1,3,4,8,9,11 assigned three per group, the observed treatment group is 8,9,11. Find the exact two-sided randomization fraction at least as extreme as the observed mean difference. Justify the tail count.Written reasoning: review required
• A classroom volunteers for a randomized training experiment. A small simulated null-tail fraction is reported as proof that the training works for every school. Repair the conclusion and distinguish random assignment from random sampling.Written reasoning: review required
• A matched-pair experiment assigns one treated unit within each pair. A simulator freely chooses half of all units as treated. Explain why its reference distribution is wrong and specify the repair.Written reasoning: review required

Produce and defend a statistical investigation — Statistical investigations, simulation and probability decisions

• Use RandomizationExplorer with a prespecified two-sided rule and 200 draws. Retain observed assignment, three simulated assignments/statistics, seed, exact tail probability and empirical tail fraction. Separately find the exact two-sided fraction for fixed outcomes 0,2,5,7,9,12 when treatment receives 7,9,12 and control 0,2,5. Explain which causal statement needs randomized assignment.Written reasoning: review required

Decide what the evidence can support next — Statistics foundations: data, design and sampling variability

• A voluntary poll of10,000 visitors yields a tiny reported standard error. Audit a claim about all city residents.Written reasoning: review required

Separate detectable differences from useful changes — Statistical inference: estimates, tests and defensible conclusions

• An estimated time saving has interval [0.2,0.8] seconds. Deployment requires at least 2 seconds of mean saving. Interpret the evidence without using a binary significance label alone.Written reasoning: review required

Inference: choose methods and defend conclusions — Statistical inference: estimates, tests and defensible conclusions

• A web poll of 10,000 volunteers reports 62% support and a very narrow nominal normal interval. A different study randomly assigns 80 volunteers to two teaching methods and measures exam scores. Write a short decision brief: what can each study support, what must be checked, and what additional design would support generalization to all students?Written reasoning: review required

Build a coherent probability model — Statistics foundations: data, design and sampling variability

• For a fair six-sided die, find the probability of a value that is even or greater than4.Automatically checked response; reasoning still matters
• P(A)=2/5, P(B)=1/2 and P(A and B)=1/5. Find P(A or B).Automatically checked response; reasoning still matters
• Three independent fair coin flips are made. Find the probability of at least one head.Automatically checked response; reasoning still matters
• Can P(A)=0.7, P(B)=0.6 and P(A and B)=0.1 describe one probability model? Explain.Written reasoning: review required

Restrict the reference set before multiplying — Statistics foundations: data, design and sampling variability

• A table gives P(A)=0.4, P(B)=0.5, and P(A and B)=0.2. Check independence and explain why this is a model statement, not proof of population independence from a small sample.Written reasoning: review required

Change the denominator deliberately — Statistics foundations: data, design and sampling variability

• A succeeds 18/30 of the time and B succeeds 42/70. In this table:Automatically checked response; reasoning still matters

Restrict the reference set before multiplying — Statistics foundations: data, design and sampling variability

• P(A and B)=3/20 and P(B)=1/4. Find P(A given B).Automatically checked response; reasoning still matters
• A table gives P(A)=0.4, P(B)=0.5, and P(A and B)=0.2. Check independence and explain why this is a model statement, not proof of population independence from a small sample.Written reasoning: review required
• If P(B)=0, the elementary ratio definition P(A given B) is:Automatically checked response; reasoning still matters
• Give an example where P(A given B) differs from P(B given A), and calculate both.Written reasoning: review required

Change the denominator deliberately — Statistics foundations: data, design and sampling variability

• Construct a two-way table: 40 cyclists include 12 helmet nonusers; 60 walkers include 9 hat nonusers. Explain why “helmet versus hat use” needs a measurement warning.Written reasoning: review required
• A succeeds 18/30 of the time and B succeeds 42/70. In this table:Automatically checked response; reasoning still matters

Statistics: design, variation and model selection — Statistics foundations: data, design and sampling variability

• Among 80 morning commuters, 48 use the bus; among 120 afternoon commuters, 42 use it. Construct the full two-way count table. Find P(bus), P(morning | bus), and P(bus | morning). Are time group and bus use independent in this empirical distribution? Explain why reversing a conditional is not justified.Written reasoning: review required

Update a probability without losing the base rate — Statistics foundations: data, design and sampling variability

• Factory A makes60% of items and has2% defects; B makes40% and has5% defects. Among defects, what fraction came from B? Show a frequency tree per10,000 items.Written reasoning: review required
• Explain why a positive predictive probability from one population may not transfer to another even if the test mechanism is unchanged.Written reasoning: review required

Restrict the reference set before multiplying — Statistics foundations: data, design and sampling variability

• Give an example where P(A given B) differs from P(B given A), and calculate both.Written reasoning: review required

Change the denominator deliberately — Statistics foundations: data, design and sampling variability

• Group A has 18 successes and 12 failures. What is the success proportion within A?Automatically checked response; reasoning still matters
• A has 18 successes; B has 42 successes. Among successes, what fraction belongs to A?Automatically checked response; reasoning still matters

Restrict the reference set before multiplying — Statistics foundations: data, design and sampling variability

• P(A and B)=3/20 and P(B)=1/4. Find P(A given B).Automatically checked response; reasoning still matters

Build a coherent probability model — Statistics foundations: data, design and sampling variability

• P(A)=2/5, P(B)=1/2 and P(A and B)=1/5. Find P(A or B).Automatically checked response; reasoning still matters
• For a fair six-sided die, find the probability of a value that is even or greater than4.Automatically checked response; reasoning still matters
• Can P(A)=0.7, P(B)=0.6 and P(A and B)=0.1 describe one probability model? Explain.Written reasoning: review required

Restrict the reference set before multiplying — Statistics foundations: data, design and sampling variability

• A bag has3 red and2 blue balls. Draw two without replacement. Find P(both red).Automatically checked response; reasoning still matters
• With the same bag, replace and remix the first ball. Find P(both red).Automatically checked response; reasoning still matters

Check trials before counting successes — Statistics foundations: data, design and sampling variability

• A bag contains3 red and2 blue balls. Explain why the red count in two draws without replacement is not Binomial(2,3/5); compare probabilities of two reds.Written reasoning: review required

Statistics: design, variation and model selection — Statistics foundations: data, design and sampling variability

• Under independent trials with constant success probability 1/4, compute the chance of exactly two successes in four trials and the chance of at least one. Then explain why drawing four objects without replacement from a box containing two successes and six failures uses a different model, and give its probability of exactly two successes.Written reasoning: review required

Decide what makes outcomes different — Discrete structures: counting, graphs and algorithms

• Assign president, secretary and treasurer to distinct people from 7. How many assignments?Automatically checked response; reasoning still matters
• Choose an unordered committee of 3 from 8 distinct people. How many?Automatically checked response; reasoning still matters
• Derive C(n,k)=n!/[k!(n−k)!] for 0≤k≤n by forgetting order.Written reasoning: review required

Count ordered and unordered selections — Statistical investigations, simulation and probability decisions

• Among six learners, two are experienced. Uniformly assign chair and recorder without replacement. What is the probability both roles go to experienced learners?Automatically checked response; reasoning still matters
• Label six learners A–F. Compare a uniform two-person committee with uniform chair/recorder selection. Give sample-space counts, the probability the selected people are exactly A,B, and a small diagram/table explaining the factor of two.Written reasoning: review required
• A box has four distinct red tokens and three distinct blue tokens. Draw three without replacement. Find the probability of exactly two red tokens using combinations, then explain an equivalent ordered approach.Written reasoning: review required
• Two names are drawn independently with replacement from five equally likely names. Find the probability the names differ, and explain why 5·4 cannot be the full denominator.Written reasoning: review required
• A fair coin chooses team A or team B, then a person is drawn uniformly from that team. Team A has two people and B has six. Is the resulting person uniform among all eight? Design a uniform alternative.Written reasoning: review required

Produce and defend a statistical investigation — Statistical investigations, simulation and probability decisions

• Seven named applicants include three with certification. Uniformly choose chair and deputy without replacement. Compute the probability both are certified and the probability the selected people are exactly two specified certified applicants in either role order. Compare an unordered two-person committee.Written reasoning: review required

Average outcomes with probability weights — Statistics foundations: data, design and sampling variability

• X takes 0, 2 and 5 with probabilities 1/2, 1/4 and 1/4. Find E(X).Automatically checked response; reasoning still matters
• Does the expected value7/4 have to be a possible observation of X?Automatically checked response; reasoning still matters

Read a distribution through its CDF — Calculus-based probability: laws, dependence and limits

• Write the full CDF for P(X=1)=1/3 and P(X=4)=2/3, and recover both masses from jumps.Written reasoning: review required

Map outcomes to a probability distribution — Statistical investigations, simulation and probability decisions

• Two independent fair coins produce X=3 for HH, X=1 for exactly one head and X=0 for TT. Construct the complete outcome mapping, probability table and mass graph, then calculate expectation.Written reasoning: review required
• Someone uses the preceding fair-coin distribution when the second coin result is forced to equal the first. Rebuild the law and explain which assumption changed.Written reasoning: review required
• A learner assigns equal probability 1/3 to the three values 0,1,3 because the variable has three possible values. Refute that reasoning using the elementary outcome table.Written reasoning: review required

Produce and defend a statistical investigation — Statistical investigations, simulation and probability decisions

• Roll a fair six-sided die and toss an independent fair coin. Let X equal the die value if heads and 0 if tails. Construct all elementary outcome mappings or a complete equivalent table, its mass graph and expectation. Explain why the seven distinct X values are not equally likely.Written reasoning: review required

Average outcomes with probability weights — Statistics foundations: data, design and sampling variability

• X takes 0, 2 and 5 with probabilities 1/2, 1/4 and 1/4. Find E(X).Automatically checked response; reasoning still matters
• A game pays 10 units with probability 1/5 and 0 otherwise, and costs 3 to enter. Compute expected net payoff and explain why it does not predict one play.Written reasoning: review required
• Two options each have expected payoff 5. One always pays 5; the other pays 0 or 10 equally. Compare what expectation omits.Written reasoning: review required

Statistics: design, variation and model selection — Statistics foundations: data, design and sampling variability

• A synthetic game pays 0, 4 or 10 tokens with probabilities 1/2, 1/3 and 1/6. It costs 3 tokens to play. Find the expected net result and its variance. Explain why neither a positive expected value nor its variance establishes that a particular person should play.Written reasoning: review required

Check trials before counting successes — Statistics foundations: data, design and sampling variability

• For4 independent trials with success probability1/2, find P(X=2).Automatically checked response; reasoning still matters
• For a binomial count with n=20,p=3/10, find its variance.Automatically checked response; reasoning still matters

Average outcomes with probability weights — Statistics foundations: data, design and sampling variability

• X takes 0, 2 and 5 with probabilities 1/2, 1/4 and 1/4. Find E(X).Automatically checked response; reasoning still matters

Statistics: design, variation and model selection — Statistics foundations: data, design and sampling variability

• A synthetic game pays 0, 4 or 10 tokens with probabilities 1/2, 1/3 and 1/6. It costs 3 tokens to play. Find the expected net result and its variance. Explain why neither a positive expected value nor its variance establishes that a particular person should play.Written reasoning: review required

Map outcomes to a probability distribution — Statistical investigations, simulation and probability decisions

• Two independent fair coins produce X=3 for HH, X=1 for exactly one head and X=0 for TT. Construct the complete outcome mapping, probability table and mass graph, then calculate expectation.Written reasoning: review required
• Someone uses the preceding fair-coin distribution when the second coin result is forced to equal the first. Rebuild the law and explain which assumption changed.Written reasoning: review required
• A game pays gross prize X from the first coin rule and charges a fee of 3/2 currency units per play. Construct the net-payoff law and explain its expected value.Written reasoning: review required

Produce and defend a statistical investigation — Statistical investigations, simulation and probability decisions

• Roll a fair six-sided die and toss an independent fair coin. Let X equal the die value if heads and 0 if tails. Construct all elementary outcome mappings or a complete equivalent table, its mass graph and expectation. Explain why the seven distinct X values are not equally likely.Written reasoning: review required

Approximate a sampling law without forgetting its model — Mathematical statistics: derivation and model criticism

• Resample two draws with replacement from observed values 0,2. What is the conditional variance of their mean?Automatically checked response; reasoning still matters
• In that bootstrap experiment, what is P*(bootstrap mean=1)?Automatically checked response; reasoning still matters
• Derive the conditional variance of the iid nonparametric bootstrap mean from n observed values.Written reasoning: review required

Build a forecast from observed frequencies — Statistical investigations, simulation and probability decisions

• Construct the empirical probability table and graph for synthetic daily repair counts 0,1,3 with frequencies 5,3,2. Estimate a future-day repair count under a stated stability assumption and explain what the estimate does not say.Written reasoning: review required
• After the preceding 10 days, an additional day has 6 repairs. Recompute the empirical mean and compare with merely averaging the old mean and 6.Written reasoning: review required
• A planner assigns cost 4 currency units per repair plus a fixed daily readiness cost 2. Using the 0,1,3 empirical law with counts 5,3,2, construct the cost law and its expected value.Written reasoning: review required
• A repair log contains only weekdays with at least one technician on site. Someone uses it to predict all calendar days. Identify the target mismatch and propose an evidence plan.Written reasoning: review required

Produce and defend a statistical investigation — Statistical investigations, simulation and probability decisions

• Synthetic daily demand values 0,2,4 occur on 6,3,3 observed days. Build an empirical distribution and graph. A service incurs fixed cost 3 plus 2 per demand unit; construct its cost law and expected cost. Explain what new data would be needed before treating this as a forecast for all seasons.Written reasoning: review required

Average outcomes with probability weights — Statistics foundations: data, design and sampling variability

• A game pays 10 units with probability 1/5 and 0 otherwise, and costs 3 to enter. Compute expected net payoff and explain why it does not predict one play.Written reasoning: review required
• Two options each have expected payoff 5. One always pays 5; the other pays 0 or 10 equally. Compare what expectation omits.Written reasoning: review required

Statistics: design, variation and model selection — Statistics foundations: data, design and sampling variability

• A synthetic game pays 0, 4 or 10 tokens with probabilities 1/2, 1/3 and 1/6. It costs 3 tokens to play. Find the expected net result and its variance. Explain why neither a positive expected value nor its variance establishes that a particular person should play.Written reasoning: review required

Average outcomes with probability weights — Statistics foundations: data, design and sampling variability

• A game pays 10 units with probability 1/5 and 0 otherwise, and costs 3 to enter. Compute expected net payoff and explain why it does not predict one play.Written reasoning: review required

Statistics: design, variation and model selection — Statistics foundations: data, design and sampling variability

• A synthetic game pays 0, 4 or 10 tokens with probabilities 1/2, 1/3 and 1/6. It costs 3 tokens to play. Find the expected net result and its variance. Explain why neither a positive expected value nor its variance establishes that a particular person should play.Written reasoning: review required

Average outcomes with probability weights — Statistics foundations: data, design and sampling variability

• Two options each have expected payoff 5. One always pays 5; the other pays 0 or 10 equally. Compare what expectation omits.Written reasoning: review required

Choose an action by posterior expected loss — Mathematical statistics: derivation and model criticism

• If acting costs c when θ=0 and not acting costs d when θ=1, derive the action rule from posterior probability q=P(θ=1|x).Written reasoning: review required

Compare uncertain strategies and consequences — Statistical investigations, simulation and probability decisions

• Compare three synthetic logistics strategies at disruption probability 0.2: A loses 8 in either state; B loses 2 normally and 20 in disruption; C loses 5 normally and 12 in disruption. Construct a table of expected losses and give a reasoned choice with a risk caveat.Written reasoning: review required
• For the same three strategies, derive their expected-loss lines in p and identify which minimizes expectation across 0 ≤ p ≤ 1, including ties.Written reasoning: review required
• Two strategies have equal expected net payoff 0: one always pays 0, the other pays −100 or +100 with equal chances. Explain why expectation does not settle a one-time decision, then state what additional evidence or preference would matter.Written reasoning: review required

Produce and defend a statistical investigation — Statistical investigations, simulation and probability decisions

• A fictional equipment policy A costs 9 in every state. B costs 3 normally and 18 in disruption; C costs 6 normally and 11 in disruption. Compare at disruption probability 0.25 and identify the expected-cost minimizing policy for all 0 ≤ p ≤ 1. If an alert changes the conditional disruption probability to 0.6, reconsider and discuss risk.Written reasoning: review required

Design and audit a fair random allocation — Statistical investigations, simulation and probability decisions

• Design a fair allocation among four named people using a fair six-sided die, with at most two attempts and an explicit no-decision outcome. Prove each person’s unconditional chance and conditional chance given allocation.Written reasoning: review required
• Someone assigns die faces 1,5 to person A;2,6 to B;3 to C;4 to D. Explain why this is not a fair four-person lottery and repair it.Written reasoning: review required
• Three people independently draw a fair bit; the highest bit wins, with tied winners resolved by fixed alphabetical order. Is the result equal-chance? Give a concrete unequal event and a symmetric repair.Written reasoning: review required
• A fair lottery selects one recipient independently on each of 12 days among four people. One receives six selections. Does this alone prove unfair implementation? Distinguish equal chances from equal workload and propose an audit.Written reasoning: review required

Produce and defend a statistical investigation — Statistical investigations, simulation and probability decisions

• Design an equal-chance choice among five people using a fair six-sided die, with at most two attempts. Include a no-decision outcome, prove each unconditional and conditional selection probability, and explain why mapping the unused face to the first person would break fairness.Written reasoning: review required

Update a probability without losing the base rate — Statistics foundations: data, design and sampling variability

• Factory A makes60% of items and has2% defects; B makes40% and has5% defects. Among defects, what fraction came from B? Show a frequency tree per10,000 items.Written reasoning: review required
• Explain why a positive predictive probability from one population may not transfer to another even if the test mechanism is unchanged.Written reasoning: review required

Plan the consequences of a decision — Statistical inference: estimates, tests and defensible conclusions

• H0 says a production change does not raise defect probability. Explain the two errors and their distinct costs.Written reasoning: review required

Choose an action by posterior expected loss — Mathematical statistics: derivation and model criticism

• If acting costs c when θ=0 and not acting costs d when θ=1, derive the action rule from posterior probability q=P(θ=1|x).Written reasoning: review required

Compare uncertain strategies and consequences — Statistical investigations, simulation and probability decisions

• Compare three synthetic logistics strategies at disruption probability 0.2: A loses 8 in either state; B loses 2 normally and 20 in disruption; C loses 5 normally and 12 in disruption. Construct a table of expected losses and give a reasoned choice with a risk caveat.Written reasoning: review required
• For the same three strategies, derive their expected-loss lines in p and identify which minimizes expectation across 0 ≤ p ≤ 1, including ties.Written reasoning: review required
• A fictional alert is followed by actual disruption in 30% of alerted cases. Acting costs 4 units regardless; not acting costs 18 if disruption occurs and 0 otherwise. Compare decisions after an alert, and explain why the population disruption rate is not the correct probability here.Written reasoning: review required
• Two strategies have equal expected net payoff 0: one always pays 0, the other pays −100 or +100 with equal chances. Explain why expectation does not settle a one-time decision, then state what additional evidence or preference would matter.Written reasoning: review required

Produce and defend a statistical investigation — Statistical investigations, simulation and probability decisions

• A fictional equipment policy A costs 9 in every state. B costs 3 normally and 18 in disruption; C costs 6 normally and 11 in disruption. Compare at disruption probability 0.25 and identify the expected-cost minimizing policy for all 0 ≤ p ≤ 1. If an alert changes the conditional disruption probability to 0.6, reconsider and discuss risk.Written reasoning: review required