Week 15 — Can Flipper Length Predict Body Mass? A Short Regression Report — Traditional alternative (no AI)

Instructors: publish this instead of the adaptive version if you prefer, and unpublish or delete the adaptive one.

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 three misreadings of a regression.

This week you learned to fit and read a least-squares line, to measure its misses, to read r² and a residual plot, and to test and estimate its slope. 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.

Correct three flawed claims

Each of these 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.

Deliverables

Submit a single regression report containing:

• the four answers, each with the arithmetic shown and a brief interpretation as the task describes;
• your corrections of the three claims, each quoting the claim, saying what is wrong with it, and rewriting it so it is correct;
• your three procedure choices, each with its reason and the alternative that fits less well;
• your message to the volunteer, with a reason tied to the penguin numbers and one named limit; and
• a one-line restatement of the values you used, so your results are reproducible.

Keep it to roughly two pages. Clear and correct beats long.

Choose how you show it. Present your report as written text, or as a recorded audio or video explanation of about three minutes 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. 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.

How you'll be graded

The full analytic rubric is attached to this assignment in Canvas.

Academic integrity and AI use

This version is designed to be completed without AI: do the predictions, the residual, the interval, the interpretations, the corrections, the procedure choices, and the writing yourself, without a chatbot or other AI tool. You may use a calculator or a spreadsheet for the arithmetic. Submitting AI-generated work as your own, or copying another student's numbers, is an integrity violation under the course policy.

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