Week 15 — Can Flipper Length Predict Body Mass? A Short Regression Report

Assignment 14 of the term, and the last one — predict from a least-squares line, measure a miss, put an interval on the slope, spot extrapolation, choose the right procedure for a new question, and correct a chatbot's misreadings of a regression.

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 4 you measured how strongly flipper length and body mass move together, r = 0.8712; this week that pattern becomes a line you can predict with and test. 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, along with the procedures from Weeks 11 to 14.

The task

The Palmer penguins field study recorded flipper length and body mass for 342 penguins. Software fit the least-squares line of body mass on flipper length. Use these values throughout:

• the sample: n = 342 penguins with both measurements; flipper lengths run from 172 mm to 231 mm;
• the least-squares line: ŷ = −5,780.8314 + 49.6856x, where x is flipper length in mm and ŷ is predicted body mass in g;
• the correlation r = 0.8712 and the coefficient of determination r² = 0.759;
• the residual standard error: 394.2782 g, about 394.28 g;
• the slope's standard error: SE(b₁) = 1.5184 g per mm, with t ≈ 32.72 on 340 degrees of freedom and a p-value far below 0.001;
• critical values: z* = 1.960 for 95%, and t* = 1.976 for 95% with 150 degrees of freedom, which the course uses as the conservative t* for 340 degrees of freedom.

Work these four problems, showing your arithmetic:

1. A field team measures a penguin with a 200 mm flipper. Using the least-squares line ŷ = −5,780.8314 + 49.6856x, what predicted body mass, in grams, does the line give, −5,780.8314 + 49.6856 × 200?
2. Suppose a field team weighs one new penguin with a 200 mm flipper at 4,500 g, and the line predicts 4,156.29 g for that flipper. What is its residual, observed minus predicted, 4,500 − 4,156.29, in grams?
3. Software reports the slope's standard error as SE(b₁) = 1.5184 g per mm on 340 degrees of freedom, and the course uses the conservative critical value t* = 1.976 for 95%. What is the margin of error for the 95% confidence interval for the slope, 1.976 × 1.5184?
4. Someone asks the line about a 300 mm flipper. Using the least-squares line ŷ = −5,780.8314 + 49.6856x, what predicted body mass, in grams, does the line give, −5,780.8314 + 49.6856 × 300?

Beside each answer, add one brief interpretation:

• Problem 1. Say in a sentence what the slope means, check whether 200 mm sits inside the observed flipper range, and name the prediction as interpolation or extrapolation. Then correct Claim 1 below.
• Problem 2. Say which side of the line this bird sits on and compare its miss with the residual standard error, the typical size of a miss. Explain in your own words what r² = 0.759 says about flipper length and body mass. Then correct Claim 2 below.
• Problem 3. Check the four conditions for regression inference out loud (linear form, independent observations, roughly normal residuals, equal spread), and say which ones the Data Lab's residual plot and histogram would check. State H₀ and Hₐ about the population slope β₁, read the reported t statistic and p-value, and give your decision at α = 0.05. Build the 95% interval for β₁ from b₁ = 49.6856 and your margin of error, and interpret it in context. Then correct Claim 3 below.
• Problem 4. Explain why the line gives no warrant for this prediction, using the observed flipper range. Then complete the procedure choices and the message below; both count with Problem 4.

Check a chatbot's answer

A classmate asked a chatbot to explain these results. Its answer sounds confident, but each of its three claims misreads a regression. Quote each claim in your report, say what is wrong with it, and rewrite it so it is correct.

• Claim 1: "The slope, 49.69, and the correlation, 0.8712, measure the same thing, so either number tells you how strongly flipper length and body mass are related."
• Claim 2: "An r² of 0.759 means about 76% of the penguins fall right on the regression line."
• Claim 3: "Because the slope's p-value is far below 0.001, growing a longer flipper makes a penguin heavier."

Use your own numbers, and the course definitions of the slope, r², and association, as evidence.

Choose the procedure

This is the last assignment of the course, so it asks you to pull the whole toolkit together. For each of these three new questions, choose one procedure you learned in Weeks 11 to 15 (a one-sample t test or interval, a two-sample t test, a one-proportion z test or interval, a two-proportion z test, or a regression line with a t test for the slope). Justify your choice from the type of variable and the question asked, and say why one plausible alternative fits less well.

• Question (i). A clinic network wants to know whether patients' mean waiting time differs between its two locations.
• Question (ii). A coffee chain wants to predict a store's daily iced-drink sales from the afternoon temperature, and to know whether the slope is real.
• Question (iii). A city wants to know whether the share of residents who bike to work differs from 10%.

Explain it to a non-expert

A volunteer at a wildlife rescue center has never studied statistics. After seeing the penguin line, the volunteer says, "This line fits so well. Can we use it to estimate the weight of any penguin that comes in, even a much bigger kind, and doesn't it prove a long flipper makes a penguin heavy?" Write a 3–5 sentence message that answers the volunteer in everyday words, gives a reason tied to the penguin numbers, and names one limit — for example, that the line was built only from flippers between 172 mm and 231 mm, or that observed data show an association, not a cause. Explain any technical term, such as "extrapolation" or "association", in everyday words. More than one good answer is possible.

For a different, ungraded example, imagine a community-garden coordinator who reads that, across past seasons, a line predicts a plot's tomato harvest from its hours of sunlight, and who asks whether the line can predict the harvest in a greenhouse with far more light than any plot in the data. Model answer: "Not reliably. The line was built from plots within a certain range of sunlight, and the greenhouse is outside that range, so the data say nothing about what happens there. Plants may stop gaining from extra light. A small trial in the greenhouse would give a better answer than stretching the line."

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

Part 1 — Student instructions

1. Draft all four answers, the four brief interpretations with your corrections of the three claims, your three procedure choices, and your message to the volunteer 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. When the coach asks you to reflect on your revision, choose one of two options of equal weight: (a) explain which change made your report more accurate and what evidence prompted it, or (b) explain which of the chatbot's three claims you might have believed before this week and what in your own work now shows it is wrong.
5. Continue until the prompt's exit condition is met.
6. 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 least-squares line, its slope and intercept, prediction and extrapolation, residuals and the residual standard error, r-squared, inference for the slope, and choosing a procedure. 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: 342 Palmer penguins with flipper length x in mm (observed from 172 mm to 231 mm) and body mass y in g. The least-squares line is y-hat = -5,780.8314 + 49.6856x. r = 0.8712, r-squared = 0.759, residual standard error = 394.2782 g, SE(b1) = 1.5184 g per mm, t = 32.72 on 340 degrees of freedom, p-value far below 0.001. Critical values: z* = 1.960 for 95%; t* = 1.976 for 95% at 150 degrees of freedom, used as the conservative t* for 340 degrees of freedom.
PROBLEM P1: A field team measures a penguin with a 200 mm flipper. Using the least-squares line ŷ = −5,780.8314 + 49.6856x, what predicted body mass, in grams, does the line give, −5,780.8314 + 49.6856 × 200?
PROBLEM P2: Suppose a field team weighs one new penguin with a 200 mm flipper at 4,500 g, and the line predicts 4,156.29 g for that flipper. What is its residual, observed minus predicted, 4,500 − 4,156.29, in grams?
PROBLEM P3: Software reports the slope's standard error as SE(b₁) = 1.5184 g per mm on 340 degrees of freedom, and the course uses the conservative critical value t* = 1.976 for 95%. What is the margin of error for the 95% confidence interval for the slope, 1.976 × 1.5184?
PROBLEM P4: Someone asks the line about a 300 mm flipper. Using the least-squares line ŷ = −5,780.8314 + 49.6856x, what predicted body mass, in grams, does the line give, −5,780.8314 + 49.6856 × 300?
After the four answers, I must briefly interpret each one. For P1: what the slope means in a sentence, whether 200 mm is inside the observed range (interpolation), and a correction of Claim 1. For P2: which side of the line the bird sits on, a comparison with the residual standard error, what r-squared = 0.759 means, and a correction of Claim 2. For P3: the four conditions for regression inference, H0: beta1 = 0 and Ha: beta1 is not 0, the decision at alpha = 0.05 from the reported t and p-value, the 95% interval for beta1 built from b1 = 49.6856 and my margin, its interpretation, and a correction of Claim 3. For P4: why the 300 mm prediction is extrapolation, then my procedure choices and my message, which both count with P4. The three claims I must correct are: Claim 1, "The slope, 49.69, and the correlation, 0.8712, measure the same thing, so either number tells you how strongly flipper length and body mass are related." Claim 2, "An r-squared of 0.759 means about 76% of the penguins fall right on the regression line." Claim 3, "Because the slope's p-value is far below 0.001, growing a longer flipper makes a penguin heavier." The procedure choices: for each of three new questions I choose one procedure from Weeks 11 to 15 (one-sample t, two-sample t, one-proportion z, two-proportion z, or a regression line with a t test for the slope), justify it from the type of variable and the question, and say why one plausible alternative fits less well. Question (i): does patients' mean waiting time differ between a clinic network's two locations? Question (ii): can a coffee chain predict a store's daily iced-drink sales from the afternoon temperature, and is the slope real? Question (iii): does the share of a city's residents who bike to work differ from 10%? I must then write a 3-5 sentence message to a wildlife rescue center volunteer who is new to statistics and asks whether the line can estimate the weight of any penguin, even a much bigger kind, and whether it proves a long flipper makes a penguin heavy, 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 predictions, residuals, and margins (40 points), regression reasoning, choices, and corrections (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 procedure choices and my message only with P4, not again with P1-P3.
COMPUTATION UNITS: A correct prediction, residual, or margin, including a correct rounding to two decimals and the correct sign = 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, such as a spreadsheet FORECAST or SLOPE check, or equivalent correct notation.
REASONING UNITS: For P1-P3 separately, an accurate account of the regression step the problem asks for plus its paired correction = 2; exactly one of the two = 1; neither = 0. For P1 the account is the slope read as a predicted change in grams per millimetre and the interpolation check, paired with the correction of Claim 1. For P2 the account is the residual read against the typical miss and the meaning of r-squared, paired with the correction of Claim 2. For P3 the account is the conditions, the test of the slope, and the interval, paired with the correction of Claim 3. For P4, the extrapolation explanation AND three procedure choices with reasons AND a message with a reason tied to the penguin numbers and a named limit, all accurate = 2. If not all of P4 is accurate but at least one substantive part is, give 1. No accurate reasoning = 0; naming a rule alone does not earn a unit. Reading the slope as the correlation, reading r-squared as a share of points, claiming the slope proves cause, using n - 1 degrees of freedom for the slope, or trusting a prediction outside the observed range 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 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 it; let me choose either (a) which change made my report more accurate and what evidence prompted it, or (b) which of the three claims I might have believed before this week and what in my work now shows it is wrong. 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 corrected all three claims AND made the three procedure choices AND written my message to the volunteer 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 regression report, and the share link.
COPY EVERYTHING ABOVE THIS LINE

Deliverables

Submit these three items in the assignment entry:

• your revised regression report with all four answers, their brief interpretations, your corrections of the three claims, your three procedure choices, and your message to the volunteer — written, or recorded as audio or video with a transcript, or as a labelled sketch of the penguin line with the observed flipper range, your 200 mm and 300 mm predictions, and your residual marked, plus 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.

Looking ahead to the final. The final exam comes in Week 16, finals week. To prepare, use the Final study guide, the Final Practice Exam, and the exam-prep tutorial in the finals-week module. The choose-the-procedure skill you practise here, matching a question to a method and checking its conditions, is the skill those materials ask you to use across the whole course.

Equivalent route (no AI required)

This route assesses the same Week 15 skills — predicting from a least-squares line and reading its slope, computing and reading a residual, checking the conditions and putting an interval on the slope, recognizing extrapolation, choosing a procedure for a new question, and correcting misreadings of a regression — 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, the corrections of the three claims, the three procedure choices, and the message to the volunteer. 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 and the evidence that prompted the change, and then answers one of two options of equal weight: (a) which change made your report more accurate and why, or (b) which of the three claims you might have believed before this week and what in your work now shows it is wrong. Submit the revised 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, corrections, procedure choices, 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