Week 9 — How Much Slack Should the Timetable Allow? A Short Normal-Model Report — Traditional alternative (no AI)

Instructors: publish this instead of the adaptive version if you prefer, and unpublish or delete the adaptive one.

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 week you learned the normal model, the empirical rule, standardizing with z, and forward and inverse normal calculations, and you learned to check whether a normal model is reasonable before you lean on it. Last week you met the z-score as relative standing; this week it gets a curve to live on. In this assignment you work four short problems, interpret each one, and explain a percentile to someone who has never studied statistics. This week's lecture tutorial, quiz, discussion, and Data Lab practise the same ideas from other angles.

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.

Show your arithmetic. A reader should be able to reproduce every number you report from the values above without guessing which values or steps you used.

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.

Deliverables

Submit a single normal-model report containing:

• the four answers, each with the arithmetic shown and a brief interpretation of which procedure you used, what the value means, and one use or limit of it;
• your verdict on the penguin body masses, reasonable or not reasonable, with at least one named check;
• your message to the carpool organizer, explaining how early to leave with a reason tied to the commute model's numbers and naming one limit; and
• a one-line restatement of the values you used, so your results are reproducible.

Keep it to roughly one page. Clear and correct beats long.

Choose how you show it. Present your report as written text, or as a recorded audio or video explanation of about three minutes 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. 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.

How you'll be graded

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

Academic integrity and AI use

This version is designed to be completed without AI: do the standardizing, the table lookups, the interpretations, and the writing yourself, without a chatbot or other AI tool. You may use a calculator, a printed standard normal table, or a spreadsheet's NORM.DIST and NORM.INV for the arithmetic. Submitting AI-generated work as your own, or copying another student's numbers, is an integrity violation under the course policy.

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