Week 12 — How Far Can a Share Be Trusted? A Short Proportion Report

Assignment 11 of the term — find the margin of error for a share, build and interpret a one-proportion z interval, read a poll's margin in percentage points, choose a sample size, and explain a poll's plus-or-minus 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 learned that voluntary-response and convenience samples build in bias, in Week 10 you saw how a sample proportion wobbles, and in Week 11 you built your first confidence interval; this week those ideas meet in the confidence interval for a proportion. 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

Of the 333 penguins in the Palmer field study with a recorded sex, 165 are female and 168 are male, so the sample proportion of females is p̂ = 165 ⁄ 333 ≈ 0.4955, and 1 − p̂ = 0.5045. The standard error of p̂ is SE = √(p̂(1 − p̂) ⁄ n) ≈ 0.0274, and the 95% critical value is z* = 1.96. A pollster who has no earlier estimate plans with the conservative p̂ = 0.5.

Work these four problems, showing your arithmetic:

1. Of the 333 Palmer penguins with a recorded sex, 165 are female, so p̂ = 0.4955, and the standard error of p̂ is SE = 0.0274. With the 95% critical value z* = 1.96, what is the margin of error, z* × SE?
2. The penguin share is p̂ = 0.4955 and its 95% margin of error is 0.0537. What is the lower end of the 95% one-proportion z interval, p̂ − margin?
3. A pollster plans with the conservative p̂ = 0.5 and surveys n = 100 people, so the standard error is √(0.5 × 0.5 ⁄ 100) = 0.05. With z* = 1.96, what is the 95% margin of error, z* × SE?
4. A city wants a 95% margin of error of plus or minus 2 percentage points, ME = 0.02, and plans with the conservative p̂ = 0.5, so p̂(1 − p̂) = 0.25. Using n = (z* ⁄ ME)² × p̂(1 − p̂) with z* = 1.96, how many people should it survey?

Beside each answer, add one brief interpretation: the idea behind the value, what the value means, and one check, use, or limit. For problem 1, check that 165 females and 168 males are both at least 10, and write the margin in percentage points. For problem 2, also find the upper end, 0.4955 + 0.0537, write the interpretation sentence "we are 95% confident that the population proportion of female penguins is between … and …", and say whether an even split, 0.5, is plausible. For problem 3, write the margin in percentage points, say why 0.5 is the conservative choice, and say what happens to the margin with four times the people. For problem 4, say why a sample size is always rounded up, compare your answer with the 1,068 people needed for plus or minus 3 percentage points, and name one kind of bias the margin of error does not cover.

Explain it to a non-expert

A member of a city council has never studied statistics. A news story says, "An online survey found that 52% of residents support a new bus line, with a margin of error of plus or minus 3 percentage points." The survey was a link that readers chose to click. The council member asks, "So support is definitely between 49% and 55%, right?" Write a 3–5 sentence message that answers in everyday words, gives a reason tied to what the margin of error measures, and names one limit — for example, that people who choose to answer an online survey may differ from everyone else, or that the wording of the question can push the answers. Explain any technical term, such as "margin of error" or "percentage point", in everyday words. More than one good answer is possible.

For a different, ungraded example, imagine a gym manager who reads that a random sample of members found 30% want more evening classes, plus or minus 4 percentage points. Model answer: "The plus-or-minus tells you how much that share would bounce around if you asked a different random group of members, so the true share is plausibly anywhere from about 26% to 34%. That range only covers luck in who was picked. If the survey went out by email and mostly regulars answered, the share could be off by more than 4 percentage points, and the plus-or-minus would not show it."

Use that pattern for the council member's question, with your own explanation and evidence.

Part 1 — Student instructions

1. Draft all four answers, the four brief interpretations, and your message to the council member 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 standard error of a sample proportion, the conditions for a one-proportion z interval, building and interpreting the interval, a poll's margin of error in percentage points, choosing a sample size, and what the margin of error does not cover. 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: of 333 Palmer penguins with a recorded sex, 165 are female and 168 are male, so p-hat = 0.4955 and 1 - p-hat = 0.5045; the standard error of p-hat is sqrt(0.4955 x 0.5045 / 333) = 0.0274 (rounded); the 95% critical value is z* = 1.96; a pollster with no earlier estimate plans with the conservative p-hat = 0.5.
PROBLEM P1: Of the 333 Palmer penguins with a recorded sex, 165 are female, so p̂ = 0.4955, and the standard error of p̂ is SE = 0.0274. With the 95% critical value z* = 1.96, what is the margin of error, z* × SE?
PROBLEM P2: The penguin share is p̂ = 0.4955 and its 95% margin of error is 0.0537. What is the lower end of the 95% one-proportion z interval, p̂ − margin?
PROBLEM P3: A pollster plans with the conservative p̂ = 0.5 and surveys n = 100 people, so the standard error is √(0.5 × 0.5 ⁄ 100) = 0.05. With z* = 1.96, what is the 95% margin of error, z* × SE?
PROBLEM P4: A city wants a 95% margin of error of plus or minus 2 percentage points, ME = 0.02, and plans with the conservative p̂ = 0.5, so p̂(1 − p̂) = 0.25. Using n = (z* ⁄ ME)² × p̂(1 − p̂) with z* = 1.96, how many people should it survey?
After the four answers, I must briefly interpret each one: the idea behind the value, what it means, and one check, use, or limit; for P1, check that 165 and 168 are both at least 10 and write the margin in percentage points; for P2, also find the upper end 0.4955 + 0.0537, write "we are 95% confident that the population proportion of female penguins is between ... and ...", and say whether 0.5 is plausible; for P3, write the margin in percentage points, say why 0.5 is the conservative choice, and say what four times the people does to the margin; for P4, say why a sample size is always rounded up, compare with the 1,068 people needed for plus or minus 3 percentage points, and name one bias the margin does not cover. I must then write a 3-5 sentence message to a city council member who is new to statistics and asks whether an online survey's 52% support, plus or minus 3 percentage points, means support is definitely between 49% and 55%, giving a reason tied to what the margin measures and naming one limit. I must explain technical terms in everyday words.

SCORING CRITERIA: Score only evidence I supply: correct proportion-interval computations (40), proportion 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 margin of error, interval end, poll margin, or sample size = 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 percentage or a reasonable rounding.
REASONING UNITS: For P1-P3 separately, an accurate account of the idea behind the value and what it means plus a check, use, or limit tied to the penguins or the poll = 2; exactly one of the two = 1; neither = 0. For P4, BOTH the rounding-up rule with a comparison to the 1,068 people needed for plus or minus 3 percentage points and a named bias the margin does not cover AND a final message with a reason tied to the margin 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. Saying there is a 95% probability that p is in one interval, reading plus or minus 3 percentage points as 3% of the share, or saying the margin of error covers bias 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 council member 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 proportion report, and the share link.
COPY EVERYTHING ABOVE THIS LINE

Deliverables

Submit these three items in the assignment entry:

• your revised proportion report with all four answers, their brief interpretations, and your message to the council member — written, or recorded as audio or video with a transcript, or as a labelled sketch of the interval on a number line from 0 to 1 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 12 skills — finding the margin of error for a share, checking the large-counts condition, building and interpreting a one-proportion z interval, reading a poll's margin in percentage points, choosing a sample size, and naming the kinds of bias a margin of error does not cover — 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 council member. 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 proportion 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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