Week 14 — Is the Gap Real? Testing Claims About Penguin Means and Shares — Traditional alternative (no AI)

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

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 week you learned to run hypothesis tests for means and proportions: the one-sample t test, the two-sample t test read from software output, the one-proportion z test, and the two-proportion z test with a pooled proportion. You also learned that a two-sided test and its confidence interval are two views of the same question, and that "not significant" is not "no difference". In Week 13 you learned the logic of a test on a guessing quiz; this week that logic goes to work on penguin means and shares. In this assignment you work four short problems, interpret each one, and explain a "not significant" result to someone who has never studied statistics. This week's lecture tutorial, quiz, discussion, and Data Lab practise the same ideas from other angles.

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.

Show your arithmetic. A reader should be able to reproduce every number you report from the values above without guessing which values or steps you used.

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.

Deliverables

Submit a single test report containing:

• the four answers, each with the arithmetic shown and a brief interpretation of the test behind the value, its decision in context, and one use or limit;
• your message to the guide, explaining what the "not significant" sex test does and does not say about these penguins, 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 one page. 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 Gentoo-minus-Adelie interval on a number line with 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.

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 gap, the expected count, the difference in proportions, the interval end, the interpretations, 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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