Week 12 · Assignment: Traditional alternative (no AI)

Week 12 — How Far Can a Share Be Trusted? A Short Proportion Report — Traditional alternative (no AI)

Introduction to Statistics Free sample edition

Traditional alternative (no AI), unpublished. In the course this is a separate assignment that ships unpublished beside the adaptive assignment: same objective, no AI. An instructor who prefers it publishes it and unpublishes or deletes the adaptive one, as the note at its top says.

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

Assignment 11 of the term — find the margin of error for a share, build and interpret a one-proportion z interval, read a poll's margin in percentage points, choose a sample size, and explain a poll's plus-or-minus to someone new to statistics.

This week you learned to build a confidence interval for a proportion: the standard error of p̂, the large-counts check, the one-proportion z interval and its interpretation, a poll's margin of error in percentage points, and how to choose a sample size. In Week 1 you learned that voluntary-response and convenience samples build in bias, in Week 10 you saw how a sample proportion wobbles, and in Week 11 you built your first confidence interval; this week those ideas meet. In this assignment you work four short problems, interpret each one, and explain a poll's plus-or-minus 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

Of the 333 penguins in the Palmer field study with a recorded sex, 165 are female and 168 are male, so the sample proportion of females is p̂ = 165 ⁄ 333 ≈ 0.4955, and 1 − p̂ = 0.5045. The standard error of p̂ is SE = √(p̂(1 − p̂) ⁄ n) ≈ 0.0274, and the 95% critical value is z* = 1.96. A pollster who has no earlier estimate plans with the conservative p̂ = 0.5.

Work these four problems, showing your arithmetic:

Of the 333 Palmer penguins with a recorded sex, 165 are female, so p̂ = 0.4955, and the standard error of p̂ is SE = 0.0274. With the 95% critical value z* = 1.96, what is the margin of error, z* × SE?
The penguin share is p̂ = 0.4955 and its 95% margin of error is 0.0537. What is the lower end of the 95% one-proportion z interval, p̂ − margin?
A pollster plans with the conservative p̂ = 0.5 and surveys n = 100 people, so the standard error is √(0.5 × 0.5 ⁄ 100) = 0.05. With z* = 1.96, what is the 95% margin of error, z* × SE?
A city wants a 95% margin of error of plus or minus 2 percentage points, ME = 0.02, and plans with the conservative p̂ = 0.5, so p̂(1 − p̂) = 0.25. Using n = (z* ⁄ ME)² × p̂(1 − p̂) with z* = 1.96, how many people should it survey?

Beside each answer, add one brief interpretation: the idea behind the value, what the value means, and one check, use, or limit. For problem 1, check that 165 females and 168 males are both at least 10, and write the margin in percentage points. For problem 2, also find the upper end, 0.4955 + 0.0537, write the interpretation sentence "we are 95% confident that the population proportion of female penguins is between … and …", and say whether an even split, 0.5, is plausible. For problem 3, write the margin in percentage points, say why 0.5 is the conservative choice, and say what happens to the margin with four times the people. For problem 4, say why a sample size is always rounded up, compare your answer with the 1,068 people needed for plus or minus 3 percentage points, and name one kind of bias the margin of error does not cover.

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 member of a city council has never studied statistics. A news story says, "An online survey found that 52% of residents support a new bus line, with a margin of error of plus or minus 3 percentage points." The survey was a link that readers chose to click. The council member asks, "So support is definitely between 49% and 55%, right?" Write a 3–5 sentence message that answers in everyday words, gives a reason tied to what the margin of error measures, and names one limit — for example, that people who choose to answer an online survey may differ from everyone else, or that the wording of the question can push the answers. Explain any technical term, such as "margin of error" or "percentage point", in everyday words. More than one good answer is possible.

For a different, ungraded example, imagine a gym manager who reads that a random sample of members found 30% want more evening classes, plus or minus 4 percentage points. Model answer: "The plus-or-minus tells you how much that share would bounce around if you asked a different random group of members, so the true share is plausibly anywhere from about 26% to 34%. That range only covers luck in who was picked. If the survey went out by email and mostly regulars answered, the share could be off by more than 4 percentage points, and the plus-or-minus would not show it."

Use that pattern for the council member's question, with your own explanation and evidence.

Deliverables

Submit a single proportion report containing:

the four answers, each with the arithmetic shown and a brief interpretation of the idea behind the value, what it means, and one check, use, or limit;
your message to the council member, explaining whether 52% plus or minus 3 percentage points means support is definitely between 49% and 55%, with a reason tied to what the margin measures 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 from 0 to 1 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 margins, the interval, the poll margin, the sample size, 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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