Week 7 — Could a Guesser Pass? A Short Binomial Report — Traditional alternative (no AI)

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

Assignment 7 of the term — find binomial chances from the guessing-quiz table, use the complement and the binomial mean, decide when a normal curve may stand in, and explain a guessing score to someone new to statistics.

This week you learned to recognize a binomial setting, to find binomial chances by formula and by technology, to compute the binomial mean, and to decide when a normal curve may stand in for a count. In Week 5 you multiplied the chances of independent events, and in Week 6 you found the mean of a random variable; this week those tools build the binomial model. In this assignment you work four short problems, interpret each one, and explain a guessing score 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 student guesses at random on a 10-question multiple-choice quiz with four options per question. Each guess is right with chance p = 0.25, the guesses are independent, and X, the number of correct answers, is binomial with n = 10. The binomial table gives the chance of each score, rounded to four decimals: P(X = 0) = 0.0563, P(X = 1) = 0.1877, P(X = 2) = 0.2816, P(X = 3) = 0.2503, P(X = 4) = 0.1460, P(X = 5) = 0.0584, P(X = 6) = 0.0162, P(X = 7) = 0.0031, P(X = 8) = 0.0004, and 0.0000 for 9 and 10. The binomial standard deviation is σ = √(10 × 0.25 × 0.75) ≈ 1.37 correct answers, and for a 100-question version the standard deviation is about 4.33.

Work these four problems, showing your arithmetic:

1. A student guesses at random on a 10-question quiz with four options per question. The binomial table gives P(X = 0) = 0.0563, P(X = 1) = 0.1877, and P(X = 2) = 0.2816. What is P(X ≤ 2), the chance the guesser gets two or fewer questions right?
2. For the same quiz, a spreadsheet's =BINOM.DIST(4, 10, 0.25, TRUE) returns P(X ≤ 4) = 0.9219. What is P(X ≥ 5), the chance the guesser scores 5 or more?
3. For the same 10-question quiz with p = 0.25, what is the binomial mean μ = np, the long-run average number of correct answers?
4. Now the guessing happens on a 100-question quiz with four options per question, so p = 0.25. What is the expected number of wrong answers, n(1 − p)?

Beside each answer, add one brief interpretation: which part of the binomial model you used, what the value means for a guesser, and one use or limit of it. For problem 3, use the standard deviation of about 1.37 to say how far a typical guessed score lands from the mean. For problem 4, check the large-counts rule of thumb for both the 10-question quiz (np = 10 × 0.25 = 2.5) and the 100-question quiz (np = 25 and your n(1 − p)), say whether a normal curve may stand in for each, and state the curve that could stand in for the 100-question count as a model you assume, with its mean and standard deviation.

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 tutoring-center coordinator has never studied statistics. A student scored 5 out of 10 on a four-option quiz and says every answer was a guess. The coordinator asks, "Is a score like that surprising for someone who only guessed?" Write a 3–5 sentence message that answers in everyday words, gives a reason tied to the guessing numbers, and names one limit — for example, that the model assumes every answer was a pure, independent guess with a 1-in-4 chance, or that a rare result can still happen by luck and does not prove anything about one student. Explain any technical term, such as "binomial" or "at least", in everyday words. More than one good answer is possible.

For a different, ungraded example, imagine a bakery manager who sees an unusually high number of burnt loaves in one day. Model answer: "If every loaf has the same small chance of burning and one loaf does not affect another, a day like this is uncommon but can still happen now and then by bad luck. So one bad day alone does not prove the oven is broken. If bad days keep coming, though, the 'same chance every time' idea is probably wrong, and it is worth checking the oven."

Use that pattern for the guessing student, with your own explanation and evidence.

Deliverables

Submit a single binomial report containing:

• the four answers, each with the arithmetic shown and a brief interpretation of which part of the binomial model you used, what the value means, and one use or limit of it;
• your message to the coordinator, explaining whether a score of 5 out of 10 is surprising for a pure guesser, with a reason tied to the guessing 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 binomial table 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 probabilities, the mean, the rule-of-thumb 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.

Built with The Course Maker · samples.thecoursemaker.com