Week 10 — How Much Does an Average Wobble? A Short Sampling Report — Traditional alternative (no AI)

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

Assignment 9 of the term — find the spread of a sample mean, keep the standard deviation and the standard error apart, judge whether one average is surprising, check a poll's sampling distribution, and explain a surprising average to someone new to statistics.

This week you learned that a statistic changes from sample to sample, that the sampling distribution of a sample mean has mean μ and standard deviation σ ⁄ √n, that the Central Limit Theorem makes averages of large samples approximately normal, and that a sample proportion has its own sampling distribution. In Week 1 you separated a parameter from a statistic, and in Week 9 you used the normal model and the empirical rule; this week those ideas meet. In this assignment you work four short problems, interpret each one, and explain a surprising average 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

One roll of a fair die takes the values 1 to 6, each with chance 1 ⁄ 6, so its population mean is μ = 3.5 and its population standard deviation is σ = 1.7078. The sum of two dice has standard deviation 2.4152. Averages of 30 rolls have mean 3.5 and standard error σ ⁄ √30 ≈ 0.3118. Under Week 9's commute model, suppose single trip times follow a normal model with μ = 32 minutes and σ = 6 minutes. A poll asks a random sample about a proposal that, suppose, 50% of a large population favors.

Work these four problems, showing your arithmetic:

1. The sum of two fair dice has standard deviation 2.4152, and the average of two rolls is that sum divided by 2. What is the standard deviation of the average of two rolls?
2. Suppose single commute times follow a normal model with μ = 32 minutes and σ = 6 minutes. What is the standard error of the average of n = 4 trips, σ ⁄ √n, where √4 = 2?
3. Averages of 30 rolls of a fair die have mean μ = 3.5 and standard error 0.3118. What is μ − 2 × SE, the lower end of the range that holds about 95% of 30-roll averages?
4. Suppose 50% of a large population favors a proposal, so p = 0.5, and a pollster surveys n = 1,000 people at random. What is p(1 − p) ⁄ n, the quantity under the square root in the standard deviation of p̂?

Beside each answer, add one brief interpretation: which sampling idea you used, what the value means, and one use or limit of it. For problem 1, also check your answer against the rule σ ⁄ √n with σ = 1.7078 and n = 2. For problem 2, say which number describes one trip and which describes the average of 4 trips. For problem 3, also find the upper end, μ + 2 × SE, and say which 30-roll averages would be surprising for a fair die. For problem 4, take the square root to get the standard deviation of p̂, say how close to 50% about 95% of such polls land, and check np and n(1 − p) against 10 before stating a normal model for p̂ as an assumption.

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

The owner of a board-game café has never studied statistics. To test a new die, the owner rolled it 30 times and got an average of 4.6, and asks, "Is this die unfair, or was that only luck?" Write a 3–5 sentence message that answers in everyday words, gives a reason tied to the range that holds about 95% of 30-roll averages for a fair die, and names one limit — for example, that a fair die still lands outside that range now and then, or that the test assumes the rolls were fair and did not affect one another. Explain any technical term, such as "average" or "standard error", in everyday words. More than one good answer is possible.

For a different, ungraded example, imagine a library manager who sees that the average number of books borrowed per visitor was unusually high in one week. Model answer: "Weekly averages always bounce around a little, because each week brings a different mix of visitors. A small jump is the kind of change you would expect by luck, but a jump far outside the usual range is harder to explain that way. Before deciding anything changed, I would compare this week with the typical week-to-week wobble and check whether the week was unusual, such as a school holiday."

Use that pattern for the café owner's die, with your own explanation and evidence.

Deliverables

Submit a single sampling report containing:

• the four answers, each with the arithmetic shown and a brief interpretation of which sampling idea you used, what the value means, and one use or limit of it;
• your message to the café owner, explaining whether an average of 4.6 in 30 rolls makes the die look unfair, with a reason tied to the 30-roll range 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 sampling distributions 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 spreads, the standard errors, the range, the poll check, 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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