Week 9 — How Much Slack Should the Timetable Allow? A Short Normal-Model Report

Assignment 8 of the term — use the empirical rule, standardize with z, run forward and inverse normal calculations, judge whether a normal model is reasonable, and explain a percentile to someone new to statistics.

This adaptive assignment uses a coaching dialogue to help you turn a first attempt into a clear, evidence-based report. You remain the author. The coach asks questions, checks the reasoning you show, and waits for you to revise before it produces the required report. Last week you met the z-score as relative standing; this week the z-score gets a curve, and you use it to say how common a commute time is and how much slack a timetable needs. This week's lecture tutorial, quiz, discussion, and Data Lab practise the same ideas from other angles; this assignment is where you put them together.

The task

The commute-time normal model is the week's shared example: suppose one-way commute times follow a normal model with mean μ = 32 minutes and standard deviation σ = 6 minutes. This is an assumption you make, not a fact about any real city. The pinned standard normal values you need are the area to the left of z = 1, which is 0.8413, and the z-score with 95% of the area to its left, which is 1.645. For the model check in problem 4, use the 342 measured penguin body masses from earlier weeks: their mean is 4,201.75 g, their median is 4,050 g, and the median sits 500 g above the first quartile but 700 g below the third quartile.

Work these four problems, showing your arithmetic:

1. Suppose one-way commute times follow a normal model with mean 32 minutes and standard deviation 6 minutes. About 95% of trips fall within two standard deviations of the mean. How many minutes is the upper end of that range?
2. Suppose one-way commute times follow a normal model with mean 32 minutes and standard deviation 6 minutes. What is the z-score for a 23-minute trip?
3. The standard normal table gives 0.8413 as the area to the left of z = 1, and under the commute model a 38-minute trip has z = 1. What proportion of trips take longer than 38 minutes?
4. Suppose one-way commute times follow a normal model with mean 32 minutes and standard deviation 6 minutes. The z-score with 95% of the area to its left is 1.645. What commute time, in minutes, is the 95th percentile?

Beside each answer, add one brief interpretation: which procedure you used (the empirical rule, standardizing, a forward calculation, or an inverse calculation), what the value means in minutes or as a share of trips, and one use or limit of it. For problem 4, also say whether a normal model would be reasonable for the 342 penguin body masses, naming at least one check from the values above, and give your verdict as reasonable or not reasonable.

Explain it to a non-expert

A neighbor who organizes a carpool to an early hospital shift has never studied statistics. The group keeps asking, "How early do we need to leave so we are almost never late, and do real commute times really follow a bell?" Write a 3–5 sentence message that answers in everyday words, gives a reason tied to the commute model's numbers, and names one limit — for example, that the bell is an assumption, so a few very slow days from a breakdown or a storm could make the real times lean right, or that the answer only holds while the model's mean and spread stay the same. Explain any technical term, such as "percentile" or "standard deviation", in everyday words. More than one good answer is possible.

For a different, ungraded example, imagine a café manager deciding how much milk to order each morning. Model answer: "If we treat daily milk use as a bell centered on a typical day, ordering for the 95th percentile means we run short on only about 1 day in 20. That is a safety margin, not a guarantee. The bell is our assumption, and a holiday rush or a new menu item could make real use lean higher than the model expects, so we should check last month's numbers before trusting it."

Use that pattern for the carpool, with your own explanation and evidence.

Part 1 — Student instructions

1. Draft all four answers, the four brief interpretations, and your message to the carpool organizer before opening a chatbot.
2. Open any course-approved chatbot. Copy the entire Coach Prompt box, including its first and last marker lines, and paste it as one message. The coach will ask for your draft and then work one question at a time.
3. Answer in your own words. Show calculations when asked. Use the coach's feedback to revise at least one calculation or explanation, even if your first attempt was correct.
4. Continue until the prompt's exit condition is met.
5. Save the chat's share link and copy the final report exactly as the coach gives it.

The chatbot is a coach, not a ghostwriter. It must draw out your reasoning rather than write the report for you.

Coach Prompt

COPY EVERYTHING BELOW THIS LINE
ROLE: You are a careful learning coach. Draw out and challenge MY thinking about the normal model, the empirical rule, z-scores, and forward and inverse normal calculations. Do not write my assignment for me and do not invent work I did not show.

ASSIGNMENT: Work through these four problems in order, one at a time. Do not invent, replace, or reorder a problem. The data: suppose one-way commute times follow a normal model with mean mu = 32 minutes and standard deviation sigma = 6 minutes (an assumption, not a fact about any real city); the standard normal table gives 0.8413 as the area to the left of z = 1; the z-score with 95% of the area to its left is 1.645; the 342 measured penguin body masses have mean 4,201.75 g and median 4,050 g, and the median sits 500 g above the first quartile but 700 g below the third quartile.
PROBLEM P1: Suppose one-way commute times follow a normal model with mean 32 minutes and standard deviation 6 minutes. About 95% of trips fall within two standard deviations of the mean. How many minutes is the upper end of that range?
PROBLEM P2: Suppose one-way commute times follow a normal model with mean 32 minutes and standard deviation 6 minutes. What is the z-score for a 23-minute trip?
PROBLEM P3: The standard normal table gives 0.8413 as the area to the left of z = 1, and under the commute model a 38-minute trip has z = 1. What proportion of trips take longer than 38 minutes?
PROBLEM P4: Suppose one-way commute times follow a normal model with mean 32 minutes and standard deviation 6 minutes. The z-score with 95% of the area to its left is 1.645. What commute time, in minutes, is the 95th percentile?
After the four answers, I must briefly interpret each one: which procedure I used, what the value means in minutes or as a share of trips, and one use or limit of it; for P4, I must also say whether a normal model is reasonable for the 342 penguin body masses, name at least one check, and give a verdict of reasonable or not reasonable. I must then write a 3-5 sentence message to a carpool organizer who is new to statistics, explaining how early the group should leave, giving a reason tied to the commute model's numbers, and naming one limit. I must explain technical terms in everyday words.

SCORING CRITERIA: Score only evidence I supply: correct normal-model computations (40 points), procedure choice and normal-model reasoning (32 points), clarity for a non-expert reader (16 points), and showing work so it can be checked (12 points). For each of P1-P4, give 2, 1, or 0 evidence units per criterion; one unit is worth 5 computation points, 4 reasoning points, 2 clarity points, or 1.5 showing-work points. Sum all four problems within each criterion. Count my final message only with P4, not again with P1-P3.
COMPUTATION UNITS: A correct value = 2; the right values and operation with only an arithmetic slip = 1; a missing result or the wrong values or operation = 0. Do not penalize an alternative valid method, such as a spreadsheet function, or equivalent correct notation, such as a percent instead of a proportion.
REASONING UNITS: For P1-P3 separately, an accurate account of which procedure applies and what the value means plus a use or limit tied to the commute setting = 2; exactly one of the two = 1; neither = 0. For P4, BOTH an accurate account of the inverse calculation with a reasonable-or-not verdict on the penguin body masses backed by a named check AND a final message with a model-specific reason and a named limit = 2. If full P4 evidence is absent but at least one accurate substantive part appears, give 1. No accurate reasoning = 0; naming a procedure alone does not earn a unit. Calling real data "normal" instead of giving a verdict, or reading the table's left area as the area above, does not count as accurate. An arithmetic slip alone is not a reasoning error.
CLARITY UNITS: For each problem, labeled evidence understandable to someone new to statistics, with technical terms explained = 2; intelligible evidence that needs rereading because of ambiguity or jargon = 1; absent or unintelligible evidence = 0. Include the final message with P4 if submitted. Missing reasoning alone does not lower clarity for otherwise clear submitted work.
SHOWING-WORK UNITS: For each problem, checkable values and steps = 2; a relevant but incomplete value or step = 1; a result alone or no method = 0. A visible method stays checkable despite an arithmetic slip.

Do NOT invent grading rules.

DIALOGUE RULES: Ask exactly one question per message and wait for my answer. Start by asking for my complete first attempt. Build each next question from my exact words. Probe an unsupported claim and ask me to examine its evidence. Keep the conversation focused on questions rather than completed work.

ENGAGEMENT GUARDS: A greeting, copied prompt, yes/no reply, or one-word answer is not a substantive exchange. Do not flatter or accept agreement as evidence. Praise only a concrete move I made. If evidence is missing, say what is missing and ask for it. Require a real revision plus a short reflection on why the revision is better. Do not issue the report early.

EXIT CONDITION: Continue until at least five substantive exchanges have occurred AND I have supplied all four answers and their brief interpretations AND written my message to the carpool organizer AND revised at least one calculation or explanation AND reflected on the revision. Stop only after whichever requirement happens last.

REPORT: After the exit condition is satisfied, output exactly these labelled lines in this order. Replace X with the exact earned score; it may end in .5 and must not be rounded to a whole number. Base every line only on evidence in this chat; write "Not supplied" for missing evidence and never raise the score for polished chatbot prose.
STUDENT'S SCORE: X/100
EVIDENCE FOR THE SCORE: [specific evidence from my calculations and reasoning]
REVISION MADE: [what I changed and why]
STRENGTH: [one earned strength]
NEXT STEP: [one concrete improvement]
Then tell me to submit this report, my revised normal-model report, and the share link.
COPY EVERYTHING ABOVE THIS LINE

Deliverables

Submit these three items in the assignment entry:

• your revised normal-model report with all four answers, their brief interpretations, and your message to the carpool organizer — written, or recorded as audio or video with a transcript, or as a labelled sketch of the bell with the shaded areas and a short note; every form is graded by the same scoring criteria;
• the coach's five-line report; and
• the conversation share link.

Submit in any accessible file type or as a link; a recorded response may be submitted as audio or video with a transcript, or as text.

Equivalent route (no AI required)

This route assesses the same Week 9 skills — using the empirical rule, standardizing with z, running forward and inverse normal calculations, and judging whether a normal model is reasonable — for comparable time and effort, about 60 to 90 minutes, the same as the coached route. Complete the same four problems, the four brief interpretations, and the message to the carpool organizer. Then audit your own first draft against the attached scoring criteria, criterion by criterion, and write a second draft. Add a 150–200 word revision memo that names one calculation or explanation you changed, the evidence that prompted the change, and one question you still have. Submit the revised normal-model report, your completed criteria checklist, and the revision memo in the assignment entry; together they replace the share link and the coach's report. This route uses the same scoring criteria and the same points, with no penalty for choosing it.

How you'll be graded

The full analytic rubric is attached to this assignment in Canvas.

Academic integrity and AI use

The calculations, reasoning, revision, and final report must be your own. Do not ask the chatbot to create the assignment or add evidence you did not produce. Submit an honest share link from your own dialogue. The equivalent route keeps the same workload and evidence and requires no AI account.

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