Week 14 — Is the Gap Real? Testing Claims About Penguin Means and Shares

Assignment 13 of the term — run a one-sample t test, a one-proportion z test, and a two-proportion z test on the penguin data, read a two-sample comparison through its interval, and explain a "not significant" result 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 13 you learned the logic of a hypothesis test on a guessing quiz; this week you run that logic on means and proportions and connect each test to its confidence interval. 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

All four problems use the Palmer penguins field study. The birds were observed in a field study rather than drawn at random, so treat them as representative of penguins like those studied and say so. The values you need:

• Adelie body mass: 151 birds, x̄ = 3,700.66 g, s = 458.57 g, SE = 37.32 g; the critical value t* for 95% with df = 150 is 1.976.
• Gentoo minus Adelie mean body mass: difference 1,375.35 g, standard error of the difference 58.81 g; software reports Welch t = 23.386 on about 249.6 degrees of freedom, with a p-value far below 0.001.
• Sex: 165 females among 333 sexed penguins; the test of H₀: p = 0.5 gives z = −0.164 and a two-sided p-value of about 0.87; the 95% interval for p runs from 0.4418 to 0.5492.
• Adelie counts by island: 44 of 168 on Biscoe and 56 of 124 on Dream; the pooled proportion is 100 ⁄ 292, about 0.342, and the two-proportion test gives z ≈ −3.38.
• Critical values: z* is 1.960 for 95% and 2.576 for 99%.

Work these four problems, showing your arithmetic:

1. A field team weighed 151 Adelie penguins: x̄ = 3,700.66 g. A claim says the population mean is μ₀ = 4,000 g. What is the gap x̄ − μ₀, in grams?
2. Of 333 sexed penguins, 165 are female. Under H₀: p = 0.5, how many females would you expect, n × p₀?
3. On Biscoe, about 0.262 of 168 penguins are Adelie; on Dream, about 0.452 of 124 are Adelie. What is the difference in sample proportions, Biscoe minus Dream?
4. Gentoo minus Adelie mean body mass is 1,375.35 g with standard error 58.81 g. Using the conservative critical value t* = 1.976, what is the lower end of the 95% interval, difference − t* × SE, in grams?

Beside each answer, add one brief interpretation: the test behind the value, its decision in context, and one use or limit. For problem 1, divide your gap by SE to get the t statistic, compare it with t* = 1.976, and explain why a single 4,000 g bird is ordinary while a 4,000 g mean is not. For problem 2, say what the expected counts check, then read z = −0.164 and its p-value as a decision, and explain why "fail to reject" is not "the split is exactly even". For problem 3, name the procedure and why it fits, then use the pooled proportion and z to decide at both 0.05 and 0.01. For problem 4, also find the upper end, say whether the interval contains 0 and how that matches the Welch test, and judge whether the gap is large in practice.

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, "The sex test was not significant, so the study proved that exactly half of these penguins are female, 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 other shares such as 0.47 or 0.52 are still plausible. Explain any technical term, such as "not significant" or "confidence interval", in everyday words. More than one good answer is possible.

For a different, ungraded example, imagine a bakery owner who tested whether a new oven changes the average bake time and got a result that was not significant. Model answer: "The test did not find convincing evidence that the new oven is faster or slower. That is not the same as proving the two ovens bake in exactly the same time; the trial timed only a few batches, so a small difference could still be hiding. If bake time matters a lot, time more batches before deciding."

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 the one-sample t test, the one-proportion and two-proportion z tests, reading a two-sample comparison through its interval, and why "not significant" is not "no difference". 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: 151 Adelie penguins with xbar = 3,700.66 g, s = 458.57 g, SE = 37.32 g, and t* = 1.976 for 95% at df 150; Gentoo minus Adelie difference 1,375.35 g with SE 58.81 g, Welch t = 23.386 on about 249.6 df, p-value far below 0.001; 165 females among 333 sexed penguins, z = -0.164 for H0: p = 0.5, two-sided p-value about 0.87, 95% interval 0.4418 to 0.5492; Adelie counts 44 of 168 on Biscoe and 56 of 124 on Dream, pooled proportion 100 / 292 (about 0.342), z about -3.38; z* = 1.960 for 95% and 2.576 for 99%; the birds were observed in a field study, not drawn at random.
PROBLEM P1: A field team weighed 151 Adelie penguins: x̄ = 3,700.66 g. A claim says the population mean is μ₀ = 4,000 g. What is the gap x̄ − μ₀, in grams?
PROBLEM P2: Of 333 sexed penguins, 165 are female. Under H₀: p = 0.5, how many females would you expect, n × p₀?
PROBLEM P3: On Biscoe, about 0.262 of 168 penguins are Adelie; on Dream, about 0.452 of 124 are Adelie. What is the difference in sample proportions, Biscoe minus Dream?
PROBLEM P4: Gentoo minus Adelie mean body mass is 1,375.35 g with standard error 58.81 g. Using the conservative critical value t* = 1.976, what is the lower end of the 95% interval, difference − t* × SE, in grams?
After the four answers, I must briefly interpret each one: the test behind the value, its decision in context, and one use or limit; for P1, the t statistic (gap divided by SE), its comparison with t* = 1.976, and why one 4,000 g bird is ordinary while a 4,000 g mean is not; for P2, what the expected counts check, the decision from z = -0.164 and its p-value, and why "fail to reject" is not "the split is exactly even"; for P3, the procedure and why it fits, and the decision at 0.05 and at 0.01 from the pooled proportion and z; for P4, the upper end, whether the interval contains 0 and how that matches the Welch test, and whether the gap is large in practice. I must then write a 3-5 sentence message to a nature-center volunteer guide who is new to statistics and thinks the "not significant" sex test proved that exactly half of the penguins are female, 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 test computations (40 points), test 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 gap, expected count, difference in proportions, or interval end = 2, including a correct rounding; 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 test behind the value, its statistic, and its decision in context, plus a use or limit tied to the penguin data = 2; exactly one of the two = 1; neither = 0. For P4, BOTH an accurate reading of the interval for the difference (it excludes 0, matching the Welch test, and the gap is large in practice) 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. Treating "fail to reject" as proof that p = 0.5, judging the mean with s instead of SE, putting p-hat instead of p0 in the one-proportion standard error, averaging the two island shares instead of pooling the counts, or calling a p-value the probability that H0 is true 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 test report, and the share link.
COPY EVERYTHING ABOVE THIS LINE

Deliverables

Submit these three items in the assignment entry:

• your revised test 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 Gentoo-minus-Adelie 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 14 skills — running a one-sample t test, checking the conditions for and interpreting a one-proportion z test, choosing and running a two-proportion z test, reading a two-sample comparison through its interval, and explaining why "not significant" is not "no difference" — 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 test 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 credit, 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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