Week 10 — How Much Does an Average Wobble? A Short Sampling Report

Assignment 9 of the term — find the spread of a sample mean, keep the standard deviation and the standard error apart, judge whether one average is surprising, check a poll's sampling distribution, and explain a surprising average 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. In Week 1 you separated a parameter from a statistic, and in Week 9 you used the normal model and the empirical rule; this week those ideas meet in the sampling distribution, which says how much a sample mean or a sample proportion wobbles from sample to sample. 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

One roll of a fair die takes the values 1 to 6, each with chance 1 ⁄ 6, so its population mean is μ = 3.5 and its population standard deviation is σ = 1.7078. The sum of two dice has standard deviation 2.4152. Averages of 30 rolls have mean 3.5 and standard error σ ⁄ √30 ≈ 0.3118. Under Week 9's commute model, suppose single trip times follow a normal model with μ = 32 minutes and σ = 6 minutes. A poll asks a random sample about a proposal that, suppose, 50% of a large population favors.

Work these four problems, showing your arithmetic:

1. The sum of two fair dice has standard deviation 2.4152, and the average of two rolls is that sum divided by 2. What is the standard deviation of the average of two rolls?
2. Suppose single commute times follow a normal model with μ = 32 minutes and σ = 6 minutes. What is the standard error of the average of n = 4 trips, σ ⁄ √n, where √4 = 2?
3. Averages of 30 rolls of a fair die have mean μ = 3.5 and standard error 0.3118. What is μ − 2 × SE, the lower end of the range that holds about 95% of 30-roll averages?
4. Suppose 50% of a large population favors a proposal, so p = 0.5, and a pollster surveys n = 1,000 people at random. What is p(1 − p) ⁄ n, the quantity under the square root in the standard deviation of p̂?

Beside each answer, add one brief interpretation: which sampling idea you used, what the value means, and one use or limit of it. For problem 1, also check your answer against the rule σ ⁄ √n with σ = 1.7078 and n = 2. For problem 2, say which number describes one trip and which describes the average of 4 trips. For problem 3, also find the upper end, μ + 2 × SE, and say which 30-roll averages would be surprising for a fair die. For problem 4, take the square root to get the standard deviation of p̂, say how close to 50% about 95% of such polls land, and check np and n(1 − p) against 10 before stating a normal model for p̂ as an assumption.

Explain it to a non-expert

The owner of a board-game café has never studied statistics. To test a new die, the owner rolled it 30 times and got an average of 4.6, and asks, "Is this die unfair, or was that only luck?" Write a 3–5 sentence message that answers in everyday words, gives a reason tied to the range that holds about 95% of 30-roll averages for a fair die, and names one limit — for example, that a fair die still lands outside that range now and then, or that the test assumes the rolls were fair and did not affect one another. Explain any technical term, such as "average" or "standard error", in everyday words. More than one good answer is possible.

For a different, ungraded example, imagine a library manager who sees that the average number of books borrowed per visitor was unusually high in one week. Model answer: "Weekly averages always bounce around a little, because each week brings a different mix of visitors. A small jump is the kind of change you would expect by luck, but a jump far outside the usual range is harder to explain that way. Before deciding anything changed, I would compare this week with the typical week-to-week wobble and check whether the week was unusual, such as a school holiday."

Use that pattern for the café owner's die, with your own explanation and evidence.

Part 1 — Student instructions

1. Draft all four answers, the four brief interpretations, and your message to the café owner 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 sampling variability, the sampling distribution of a sample mean, the standard error, the Central Limit Theorem, and the sampling distribution of a sample proportion. 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 roll of a fair die has population mean mu = 3.5 and population standard deviation sigma = 1.7078; the sum of two dice has standard deviation 2.4152; averages of 30 rolls have mean 3.5 and standard error 1.7078 / sqrt(30) = 0.3118 (rounded); suppose single commute times follow a normal model with mu = 32 minutes and sigma = 6 minutes; suppose 50% of a large population favors a proposal, so p = 0.5.
PROBLEM P1: The sum of two fair dice has standard deviation 2.4152, and the average of two rolls is that sum divided by 2. What is the standard deviation of the average of two rolls?
PROBLEM P2: Suppose single commute times follow a normal model with μ = 32 minutes and σ = 6 minutes. What is the standard error of the average of n = 4 trips, σ ⁄ √n, where √4 = 2?
PROBLEM P3: Averages of 30 rolls of a fair die have mean μ = 3.5 and standard error 0.3118. What is μ − 2 × SE, the lower end of the range that holds about 95% of 30-roll averages?
PROBLEM P4: Suppose 50% of a large population favors a proposal, so p = 0.5, and a pollster surveys n = 1,000 people at random. What is p(1 − p) ⁄ n, the quantity under the square root in the standard deviation of p̂?
After the four answers, I must briefly interpret each one: which sampling idea I used, what the value means, and one use or limit of it; for P1, check the answer against sigma / sqrt(2); for P2, say which number describes one trip and which describes the average of 4 trips; for P3, also find mu + 2 x SE and say which 30-roll averages would be surprising for a fair die; for P4, take the square root to get the standard deviation of p-hat, say how close to 50% about 95% of such polls land, and check np and n(1 - p) against 10 before stating a normal model as an assumption. I must then write a 3-5 sentence message to a board-game cafe owner who is new to statistics, who rolled a new die 30 times and got an average of 4.6, explaining whether the die looks unfair, giving a reason tied to the 30-roll range, and naming one limit. I must explain technical terms in everyday words.

SCORING CRITERIA: Score only evidence I supply: correct sampling-distribution computations (40), sampling reasoning and interpretation (32), clarity for a non-expert reader (16), and showing work so it can be checked (12). 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 spread, standard error, range endpoint, or poll quantity = 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, such as a fraction or a reasonable rounding.
REASONING UNITS: For P1-P3 separately, an accurate account of the sampling idea behind the value and what it means plus a use or limit tied to the die or the commute = 2; exactly one of the two = 1; neither = 0. For P4, BOTH the square root (about 0.0158) with the reading that about 95% of such polls land within about 0.03 of 50% and the check of np and n(1 - p) against 10 with the normal model stated as an assumption AND a final message with a reason tied to the 30-roll range 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 rule alone does not earn a unit. Using sigma as the spread of an average, saying the data become normal, or checking the large-counts rule on n alone 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 cafe owner 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 sampling report, and the share link.
COPY EVERYTHING ABOVE THIS LINE

Deliverables

Submit these three items in the assignment entry:

• your revised sampling report with all four answers, their brief interpretations, and your message to the café owner — written, or recorded as audio or video with a transcript, or as a labelled sketch of the sampling distributions 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 10 skills — finding the spread of a sample mean, keeping the standard deviation of single values apart from the standard error, using the standard error and the empirical rule to judge whether an average is surprising, and checking the large-counts rule of thumb before using a normal model for a sample proportion — 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 café owner. 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 sampling 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.

Built with The Course Maker · samples.thecoursemaker.com