Week 6 — What Should You Expect? A Short Random-Variable Report

Assignment 6 of the term — compute two expected values and two standard deviations from the dice and the calendar, say what each one means, and explain the idea 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 put a number on chance; this week chance sets a number, and you find where its long run settles and how far one result swings. 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 and explain them in plain words.

The task

You will work with three chance processes from this week. One fair die has the faces 1, 2, 3, 4, 5, and 6, each equally likely. The sum S of two fair dice has an expected value of 7 and a standard deviation of about 2.4152. And a month picked at random from a common year has 28, 30, or 31 days; the 12 months hold 365 days, and the number of days D has a standard deviation of about 0.8620 days.

Work these four problems, showing your arithmetic:

1. One fair die has the faces 1, 2, 3, 4, 5, and 6, and each face is equally likely. What is the expected value E(X) of the face?
2. The 12 months of a common year hold 365 days, so a month picked at random has an expected length of 365 divided by 12 days. Its length in hours is 24 times its length in days. What is its expected length in hours, E(H)?
3. The number of days D in a month picked at random from a common year has a standard deviation of about 0.8620 days. What is the standard deviation of the month's length in hours, 24 times D?
4. The sum S of two fair dice has a standard deviation of about 2.4152. The average of the two dice is S divided by 2. What is the standard deviation of that average?

Beside each answer, add one brief interpretation: what that value means for the chance process, and one use or limit of it. For problem 1, say what the expected value describes over many rolls. For problem 2, say whether any single month actually has that many hours. For problem 3, say what multiplying by 24 does to the spread. For problem 4, compare the spread of the average with the spread of one die, which is about 1.71.

Explain it to a non-expert

The volunteer coordinator at a community food bank has never studied statistics. They plan volunteer hours month by month and ask you two things: why the "average month" of 730 hours never actually shows up on the calendar, and how much a month's hours really vary. Write a 3–5 sentence message that answers both questions using your results from problems 2 and 3, gives a reason tied to the calendar, and names one limit — for example, that an expected value describes the long run rather than any one month, or that the swing in hours comes entirely from the different month lengths. Explain any technical term in everyday words. More than one way of explaining it can be good.

For a different, ungraded example, imagine a coffee-shop owner who reads that the shop's "expected daily customers" is a certain number and asks what that means. Model answer: "Expected here means the average over many days, not a promise for tomorrow. Some days will be busier and some quieter, and the standard deviation tells you how big that day-to-day swing usually is. So staff for the typical day, but keep someone on call for the busy ones. One limit: if something changes, like a new office opening nearby, last year's average may not describe next month."

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

Part 1 — Student instructions

1. Draft all four answers, the four brief interpretations, and your message to the coordinator 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 random variables, expected values, and standard deviations. 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: one fair die with faces 1, 2, 3, 4, 5, 6, each equally likely; the sum S of two fair dice, with expected value 7 and standard deviation about 2.4152; and a month picked at random from a common year, with 28, 30, or 31 days, 365 days across the 12 months, and a standard deviation of about 0.8620 days for its number of days D.
PROBLEM P1: One fair die has the faces 1, 2, 3, 4, 5, and 6, and each face is equally likely. What is the expected value E(X) of the face?
PROBLEM P2: The 12 months of a common year hold 365 days, so a month picked at random has an expected length of 365 divided by 12 days. Its length in hours is 24 times its length in days. What is its expected length in hours, E(H)?
PROBLEM P3: The number of days D in a month picked at random from a common year has a standard deviation of about 0.8620 days. What is the standard deviation of the month's length in hours, 24 times D?
PROBLEM P4: The sum S of two fair dice has a standard deviation of about 2.4152. The average of the two dice is S divided by 2. What is the standard deviation of that average?
After the four answers, I must briefly interpret each one: what that value means for the chance process and one use or limit of it; for P1, what the expected value describes over many rolls; for P2, whether any single month has that many hours; for P3, what multiplying by 24 does to the spread; for P4, how the spread of the average compares with one die's standard deviation of about 1.71. I must then write a 3-5 sentence message to a food-bank volunteer coordinator who is new to statistics, explaining why the average month of 730 hours never shows up on the calendar and how much a month's hours vary, with a reason tied to the calendar and one limit. I must explain technical terms in everyday words.

SCORING CRITERIA: Score only evidence I supply: correct random-variable computations (40 points), interpretation of center and spread (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 interpretation 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 expected value or standard deviation, including equivalent notation or sensible rounding = 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 or equivalent correct notation.
INTERPRETATION UNITS: For P1-P3 separately, an accurate account of what the value means for the chance process plus a use or limit tied to the dice or calendar data = 2; exactly one of the two = 1; neither = 0. For P1, reading the expected value as a forecast for one roll does not count. For P4, BOTH an accurate explanation of why dividing by 2 halves the spread AND a final message that explains an expected value and a standard deviation in everyday words with one limit = 2. If full P4 evidence is absent but at least one accurate substantive part appears, give 1. No accurate interpretation = 0. An arithmetic slip alone is not an interpretation 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 interpretation 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 coordinator 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 random-variable report, and the share link.
COPY EVERYTHING ABOVE THIS LINE

Deliverables

Submit these three items in the assignment entry:

• your revised random-variable report with all four answers, their brief interpretations, and your message to the coordinator — written, or recorded as audio or video with a transcript, or as a labelled sketch of each distribution with 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 6 skills — computing an expected value, predicting how rescaling changes a mean and a standard deviation, interpreting each one as a long-run average or a typical swing, and explaining the ideas in plain words — 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 coordinator. 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 random-variable 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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