Week 11 — How Heavy Is the Average Adelie? A Short Confidence-Interval Report — Traditional alternative (no AI)

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

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 week you learned to turn a point estimate into a confidence interval: to find the degrees of freedom and the critical value t*, to compute the margin of error, to check the conditions, and to say what a 95% confidence level does and does not mean. In Week 10 you learned how much a sample mean wobbles from sample to sample; this week that wobble becomes a margin of error. In this assignment you work four short problems, interpret each one, and explain the interval 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

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.

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, "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.

Deliverables

Submit a single interval report containing:

• the four answers, each with the arithmetic shown and a brief interpretation of what the value means and one use or limit of it;
• your message to the guide, explaining what the interval does and does not say about Adelie 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 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 degrees of freedom, the margins of error, the interval, 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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