Week 11 — How Heavy Is the Average Adelie? A Short Confidence-Interval Report

Assignment 10 of the term — find the degrees of freedom, the margin of error, and the ends of a 95% t interval for the Adelie mean, see how the sample size moves the margin, and explain the interval 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 10 you learned how much a sample mean wobbles from sample to sample; this week that wobble becomes a margin of error, and you build and interpret a 95% t interval for the Adelie mean. 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

A field team weighed 151 Adelie penguins in an observational study. The sample mean is x̄ = 3,700.66 g and the sample standard deviation is s = 458.57 g, so the standard error is SE = s ⁄ √n = 458.57 ⁄ √151 ≈ 37.32 g. The course's critical values for 95% confidence are t* = 2.262 with 9 degrees of freedom, 2.064 with 24, and 1.976 with 150, against z* = 1.960. The birds were weighed in a field study rather than drawn at random, so treat them as representative of Adelie penguins in that region and say so.

Work these four problems, showing your arithmetic:

1. A field team weighed 151 Adelie penguins and will build a one-sample t interval for their mean body mass. How many degrees of freedom does the interval use?
2. For the 151 Adelie penguins, the standard error is SE = 37.32 g and the critical value for 95% with df = 150 is t* = 1.976. What is the margin of error, t* × SE, in grams?
3. The 151 Adelie penguins have sample mean x̄ = 3,700.66 g, and the 95% margin of error is 73.74 g. What is the lower end of the 95% t interval, x̄ − margin of error, in grams?
4. If only 25 birds had been weighed with the same spread, the standard error would be SE = 91.71 g and the critical value for 95% with df = 24 would be t* = 2.064. What would the margin of error, t* × SE, be in grams?

Beside each answer, add one brief interpretation: what the value means for the penguin interval, and one use or limit of it. For problem 1, say what the degrees of freedom do and which t* they give. For problem 2, say what the margin of error measures and what it does not cover. For problem 3, also find the upper end, x̄ + margin of error, write the interval in the course's sentence ("we are 95% confident that the population mean body mass of Adelie penguins is between … and …"), and explain why the interval says nothing about where most single birds' weights fall. For problem 4, compare your 25-bird margin with the 73.74 g margin from 151 birds, and explain the two ways more birds shrink the margin.

Explain it to a non-expert

A volunteer guide at a nature center has never studied statistics. After reading the field team's report, the guide says, "So most Adelie penguins weigh between about 3,627 and 3,774 grams, right?" Write a 3–5 sentence message that answers the guide in everyday words, gives a reason tied to the penguin numbers, and names one limit — for example, that the birds were found in a field study rather than drawn at random, or that the margin of error covers only random wobble from sample to sample. Explain any technical term, such as "confidence interval" or "margin of error", in everyday words. More than one good answer is possible.

For a different, ungraded example, imagine a café owner who reads that the 95% confidence interval for the average wait at the counter is 4 to 6 minutes. Model answer: "That range is about the typical wait across all customers, not about each customer. Single waits vary a lot, so some people will still wait much longer than 6 minutes on a busy day. The range came from timing one sample of customers, so if they were all timed on quiet mornings, the true average could sit outside it."

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

Part 1 — Student instructions

1. Draft all four answers, the four brief interpretations, and your message to the guide 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 point estimates, the t distribution, the margin of error, and how to interpret a confidence interval for a mean. 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: a field team weighed 151 Adelie penguins; the sample mean is xbar = 3,700.66 g and the sample standard deviation is s = 458.57 g, so SE = 458.57 / sqrt(151) = 37.32 g (rounded); the course's 95% critical values are t* = 2.262 at df 9, 2.064 at df 24, 1.976 at df 150, and z* = 1.960; the birds were weighed in a field study, not drawn at random.
PROBLEM P1: A field team weighed 151 Adelie penguins and will build a one-sample t interval for their mean body mass. How many degrees of freedom does the interval use?
PROBLEM P2: For the 151 Adelie penguins, the standard error is SE = 37.32 g and the critical value for 95% with df = 150 is t* = 1.976. What is the margin of error, t* × SE, in grams?
PROBLEM P3: The 151 Adelie penguins have sample mean x̄ = 3,700.66 g, and the 95% margin of error is 73.74 g. What is the lower end of the 95% t interval, x̄ − margin of error, in grams?
PROBLEM P4: If only 25 birds had been weighed with the same spread, the standard error would be SE = 91.71 g and the critical value for 95% with df = 24 would be t* = 2.064. What would the margin of error, t* × SE, be in grams?
After the four answers, I must briefly interpret each one: what the value means for the penguin interval and one use or limit of it; for P1, what the degrees of freedom do and which t* they give; for P2, what the margin of error measures and what it does not cover; for P3, the upper end, the interval in the course sentence "we are 95% confident that the population mean body mass of Adelie penguins is between ... and ...", and why the interval says nothing about where most single birds weigh; for P4, a comparison of my 25-bird margin with the 73.74 g margin from 151 birds and the two ways more birds shrink the margin. I must then write a 3-5 sentence message to a nature-center volunteer guide who is new to statistics and thinks most Adelie penguins weigh between about 3,627 and 3,774 grams, answering in everyday words, giving a reason tied to the penguin numbers, and naming one limit. I must explain technical terms in everyday words.

SCORING CRITERIA: Score only evidence I supply: correct interval computations (40 points), interval reasoning and interpretation (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 degrees of freedom, margin of error, or interval end = 2, including a correct rounding to two decimals; 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.
REASONING UNITS: For P1-P3 separately, an accurate account of the idea behind the value and what it means plus a use or limit tied to the penguin interval = 2; exactly one of the two = 1; neither = 0. For P3 the account includes the upper end and the interpretation sentence about the population mean. For P4, BOTH a comparison of the 25-bird and 151-bird margins that explains the sample-size lever through sqrt(n) and t* AND a final message with a reason tied to the penguin numbers 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 df = n, dividing s by n instead of sqrt(n), saying there is a 95% probability that the population mean is in this one interval, or saying 95% of the penguins weigh between the two ends 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 guide 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 interval report, and the share link.
COPY EVERYTHING ABOVE THIS LINE

Deliverables

Submit these three items in the assignment entry:

• your revised interval report with all four answers, their brief interpretations, and your message to the guide — written, or recorded as audio or video with a transcript, or as a labelled sketch of the interval on a number line 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 11 skills — finding the degrees of freedom, computing the margin of error and the ends of a one-sample t interval, interpreting the interval as a statement about the population mean, and explaining how the sample size moves the margin of error — 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 guide. 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 interval 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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