Practice and capstone

Practice workbook

Attempt these 36 exercises before opening the separate solutions. Show the denominator, units and reasoning, even if you use software. Each lesson has three tasks: a calculation, an explanation and a transfer or critique. A calculator is enough for the small examples. Use the dictionary and supplied CSVs for file-based tasks.

Lesson 1

1A Define the unit

A delivery table has 100 unique shipment IDs. After joining tracking events it has 340 rows. Write a question the 340-row count answers and a question requiring 100 as the denominator. Explain why treating 340 rows as 340 independent shipments is wrong.

1B Keep unknowns visible

Six shipments have known late statuses: 1, 0, 0, 1, 0, 1. Two more have unknown timing. Calculate the observed late fraction among evaluable shipments. Write a one-sentence result including coverage. Give the possible all-eight late-rate bounds without assuming the missing statuses.

1C Write an analysis contract

For “Which service is more reliable?”, define population, period, unit, outcome, comparator and one limitation using the main shipment fixture. Explain why a comparison of actual minutes alone may not answer your question.

Lesson 2

2A Change the tail

For 8, 9, 10, 11, 12, 60 minutes, calculate mean, median, linear Q1, Q3 and IQR. Compare with the lesson's original six values. Which summaries changed?

2B Investigate a flag

An analyst deletes every duration beyond a 1.5-IQR fence and reports the reduced mean as “the true average.” Identify two problems and propose a transparent analysis policy.

2C Interpret ratings

Two groups each have mean rating 3. Group A's ratings are 3, 3, 3, 3; B's are 1, 1, 5, 5. Explain what the mean hides and choose a useful display.

Lesson 3

3A Match the question

Choose a chart for each: distribution of 240 delivery durations; late rate by service; late rate over eight ordered weeks; relationship between distance and duration. State axes and units.

3B Repair the display

A bar chart compares rates of 48% and 52% with an axis from 47% to 53%, no denominators and the title “Performance doubled.” Describe the corrected display and a defensible title.

3C Write a text alternative

Write two sentences describing Lesson 3’s illustrative hub chart for a reader who cannot see it: East has 18/120 late shipments and West has 12/120. Include rates, direction, absolute percentage-point gap, denominators and the observational limitation. These are the lesson’s toy values, not the main fixture.

Lesson 4

4A Sampling versus assignment

A company recruits 40 volunteers from one shift and randomly assigns 20 to a checklist. Which feature supports a causal comparison within the study? Which limitation affects generalization? Name a condition under which even the within-study comparison could be compromised.

4B Bound missing ratings

Eight of ten ratings are observed with sum 32; the scale is 1–5. Calculate the respondent mean and the lowest/highest possible all-ten mean. Are these bounds a confidence interval?

4C Count independent units

A trial records ten tasks from each of four people. Explain why this is different from one task each from 40 people. What design or method information would you request before computing a standard error?

Lesson 5

5A Change the base rate

Among 1,000 hypothetical jobs, 200 are delayed. An alert catches 80% of delayed jobs and flags 10% of non-delayed jobs. Complete the four-cell table and calculate the fraction of alerts that identify delayed jobs.

5B Use a complement

Under a separate independent-trial model with late probability 0.10 per delivery, calculate the probability that at least one of three deliveries is late. State the assumption needed to multiply probabilities.

5C Read the condition

Using the lesson's priority example, explain the difference between P(late given priority), P(priority given late) and P(priority and late). Give the three fractions.

Lesson 6

6A Change sample size

A sample SD is 6 minutes. Calculate the estimated SE for n=9 and n=36 independent observations, assuming SD remains 6. Explain why the individual SD did not halve.

6B Read the miniature sampling table

From the nine equally likely ordered samples in Lesson 6, calculate the probability that the sample mean equals 4 and the probability it is at least 5. Is either probability the chance that the population mean equals 4?

6C Identify false precision

A report says “We repeated each of 12 measurements ten times, so our 120-row dataset has the precision of 120 independent participants.” Identify the missing assumption and explain why additional rows may contribute less information than claimed.

Lesson 7

7A Build an interval

For an independent normal sample with n=16, mean=20 minutes and sample SD=4, use t*=2.131 to calculate a two-sided 95% mean-confidence interval. Show SE and margin.

7B Repair the interpretation

Rewrite: “There is a 95% probability that the fixed population mean lies in this calculated interval, and 95% of people finish inside it.” Distinguish the two errors.

7C Change the confidence level

For unchanged data, would a 99% interval be wider or narrower than a 95% interval? Explain the trade-off and name one uncertainty that neither automatically covers.

Lesson 8

8A Explain the trial

Use the actual trial results to write three sentences: effect and direction; interval and p-value; one model/design limitation. Do not use “proves,” “chance the null is true,” or an individual guarantee.

8B Read a non-rejection

A different trial estimates checklist−control at −1 minute with a 95% interval from −6 to +4 and p=0.68. Can you conclude equivalence or no effect? Explain a plausible benefit and harm compatible with the interval.

8C Match study to method

For each scenario, identify why the independent-groups Welch method is appropriate or incomplete: different people assigned to two groups; the same people before and after; 40 tasks performed by four randomized teams. No new method formulas are required.

Lesson 9

9A Report both scales

A late rate changes from 12% to 9%. Calculate absolute percentage-point change and relative percentage change. Explain why “down 3%” is ambiguous.

9B Standardize a mixture

Use the Lesson 9 on-time table. Calculate each hub's rate for a target population containing 80% short and 20% long routes. Compare with the equal-mixture results and explain why changing weights changes the question.

9C Apply a practical threshold

The actual trial interval is −5.85 to −0.55 minutes. A worthwhile reduction is at least 2 minutes. Explain whether the interval excludes no improvement, whether it establishes the worthwhile threshold, and what decision you recommend given the data is simulated.

Lesson 10

10A Calculate and interpret

Use fitted minutes=12.6622+1.8058×km. Calculate the fitted value at 40 km. For a hypothetical observed duration of 90 minutes, calculate the residual. Interpret its sign.

10B Critique a claim

A slide says “r=0.923 proves distance causes 92.3% of delays.” Identify at least three problems with that sentence. Rewrite it accurately.

10C Check the support

The recorded distances range from 5.2 to 54.8 km. Explain why the intercept at zero and a prediction at 100 km require caution. Name two patterns you would look for in a residual plot.

Lesson 11

11A Write for a decision

Write a 100–150-word brief recommending an operational investigation using the shipment results. Include scope, a checked finding, a confounding or missingness limitation and a next step. Do not assign causal blame.

11B Separate the roles

Label each as observation, interpretation or recommendation: “20 of 51 late shipments have missing ratings”; “Respondent ratings may overstate all-shipment satisfaction”; “Investigate response barriers before claiming overall satisfaction improved.” Explain the inference connecting them.

11C Challenge an AI summary

An AI-generated draft says: “West is clearly superior, the checklist proves a 13% productivity gain, and the 4.14 average rating represents everyone.” Identify evidence needed for each claim and rewrite the three sentences with appropriate limits.

Lesson 12

12A Pre-specify a plan

Draft the capstone analysis contract before consulting the reference answer. Identify one primary trial contrast and one exploratory shipment question. State what you will retain in an audit trail if the plan changes.

12B Trace a number

Choose one chart label from your capstone. Trace it to source file, column definitions, inclusion policy, calculation and rounding. Ask another learner to reproduce it without your help.

12C Transfer the reasoning

If tomorrow's evidence came from 40 observations on four people rather than 40 independent people, write a short note explaining which inferential claims need revision and why your descriptive arithmetic may still be correct.

Checkpoint assessments

Use closed notes first, then consult the lessons. Each checkpoint has 20 points. Answers and marking criteria are in the separate solutions page. A suggested target is 16/20 on each, followed by correction of every consequential interpretation error. These are proposed instructional thresholds, not validated examinations.

Checkpoint A After Lesson 4

A1. An order table has 80 orders; an item join has 210 rows. Define both grains and explain which denominator fits “fraction of orders delivered late.” (3 points)

A2. For 2, 3, 4, 5, 16 minutes, calculate mean and median and explain the difference. (3 points)

A3. A graph flags a 90-minute delivery as an outlier. Give two possible explanations and a defensible response. (3 points)

A4. Twenty of 50 customers respond to a survey. Explain why a 40% response rate does not establish representative satisfaction, and name a useful comparison using known data. (4 points)

A5. Distinguish random sampling from random assignment with one consequence of each. (4 points)

A6. Select a chart for distribution of durations and state the horizontal variable, vertical quantity and one chart-design risk. (3 points)

Checkpoint B After Lesson 8

B1. Out of 1,000 jobs, 50 are delayed. The alert flags 40 delayed and 95 non-delayed jobs. What fraction of flags identify delayed jobs? Explain why 40/50 answers another question. (3 points)

B2. With sample SD 8 and n=64 independent observations, calculate SE and interpret how it differs from SD. (3 points)

B3. Given mean 30, SE 2 and a supplied 95% t multiplier of 2.1, calculate and interpret a mean interval. (4 points)

B4. Explain what a two-sided p-value of 0.03 means under the null model, and give two interpretations it does not support. (4 points)

B5. Explain why a 95% interval spanning −5 to +5 minutes does not establish equivalence between two interventions. (3 points)

B6. A researcher tests 30 outcomes and reports only the smallest p-value. Explain the concern and give one design/reporting improvement. (3 points)

Checkpoint C After Lesson 11

C1. A rate falls from 10% to 8%. Give the absolute and relative change with correct units. (3 points)

C2. Explain how different route mixtures can reverse an overall hub ranking even if one hub has higher on-time rates within each route type. (3 points)

C3. A regression slope is 1.8 minutes/km. Interpret it descriptively and identify a causal claim it cannot justify on its own. (3 points)

C4. Describe two diagnostic concerns that a high R-squared does not rule out. (3 points)

C5. For the simulated trial's −3.20-minute estimate and [−5.85, −0.55] interval, compare the evidence with zero and a −2-minute worthwhile threshold. Recommend a proportionate next step. (4 points)

C6. Write a two-sentence chart caption for the trial effect, including the contrast, interval meaning, sample size and simulation limitation. (4 points)

Capstone A defensible operational decision brief

Your commission

Advise the fictional Northstar Dispatch manager on two linked decisions: what operational question to investigate next, and whether the checklist merits a real controlled field pilot. A mean task-time reduction of at least 2 minutes is the supplied hypothetical worthwhile threshold. The manager has not supplied implementation costs, safety outcomes or quality measures. Do not invent them as observed evidence.

Use the supplied 240-shipment fixture, its raw export and the separate 40-participant simulated trial. All evidence is synthetic. The goal is to show reasoning that would transfer to genuine work, without making any claim about an actual company or person.

Deliverables and stages

1. Plan: one-page analysis contract specifying population, unit, period, outcomes, contrast, missingness policy, a 2-minute practical threshold and one exploratory question. Save it before opening the reference answer.
2. Audit and calculation record: raw-to-clean reconciliation, spot-check of five records, main descriptive summaries, missing-rating counts by late status, group comparability, trial effect/CI/test and interpretation. Use calculator workings, a spreadsheet with visible formulas, or your own Python script. Preserve the original files.
3. Two original charts: one directly showing the trial difference and 95% interval against zero and the worthwhile threshold; one answering your chosen shipment question. Include underlying plotted values and a written equivalent for each.
4. Management brief: 250–400 words with recommendation, evidence, uncertainty, the most consequential limitation and a concrete next step. Include source filenames and scope without overwhelming the headline.
5. Methods appendix and review: explain assumptions, rejected/duplicate handling, quartile and SD conventions if used, rounding, exploratory choices, software versions if used and how to reproduce the result. Include a self-review and one peer challenge or your own skeptical challenge.

The optional learner_analysis.py starter and check_learner.py verify a small subset of your calculations. They are not substitutes for these deliverables. You may use the supplied distribution-function results for the trial if working without a statistical package, but must reconstruct the summaries and explain the method.

Rubric 100 points

Suggested completion target: at least 80/100 with no critical integrity failure. Score each area by the proportion of criteria convincingly met, and write one concrete improvement for every deduction. Another reviewer may reasonably differ on style points; calculations and inferential meaning should reconcile.

Critical failures requiring revision before completion: altering raw evidence to obtain preferred results; silently omitting valid records or unknowns; using duplicated/clustered rows as independent participants; claiming the simulated trial proves a real effect; causal hub attribution from the observational comparison; treating a p-value as the probability a hypothesis is true; describing a mean-confidence interval as the range containing 95% of individuals; hiding a material limitation to make the recommendation stronger.

Review questions

• Can every percentage be traced to numerator and denominator?
• Are the two files' units and study designs kept distinct?
• Would the conclusion change if the true reduction were 0.6 minutes rather than 3.2?
• Did the chart show the comparison the sentence actually makes?
• Are missing ratings and group-mix differences visible?
• What new evidence could reverse your recommendation?
• Can another learner reproduce the important numbers without asking which records you used?

The proposed rubric has not been calibrated on a learner cohort. It supports feedback and revision, not certification or a claim of professional statistical competence.