Week 7 — Could a Guesser Pass? A Short Binomial Report

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 adaptive assignment uses a coaching dialogue to help you turn a first attempt into a clear, evidence-based report. You remain the author. The coach asks questions, checks the reasoning you show, and waits for you to revise before it produces the required report. 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, and you decide when a smooth normal curve may stand in for a count. This week's lecture tutorial, quiz, discussion, and Data Lab practise the same ideas from other angles; this assignment is where you put them together.

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.

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.

Part 1 — Student instructions

1. Draft all four answers, the four brief interpretations, and your message to the coordinator before opening a chatbot.
2. Open any course-approved chatbot. Copy the entire Coach Prompt box, including its first and last marker lines, and paste it as one message. The coach will ask for your draft and then work one question at a time.
3. Answer in your own words. Show calculations when asked. Use the coach's feedback to revise at least one calculation or explanation, even if your first attempt was correct.
4. Continue until the prompt's exit condition is met.
5. Save the chat's share link and copy the final report exactly as the coach gives it.

The chatbot is a coach, not a ghostwriter. It must draw out your reasoning rather than write the report for you.

Coach Prompt

COPY EVERYTHING BELOW THIS LINE
ROLE: You are a careful learning coach. Draw out and challenge MY thinking about the binomial model, binomial probabilities, the binomial mean, and when a normal curve may stand in for a count. Do not write my assignment for me and do not invent work I did not show.

ASSIGNMENT: Work through these four problems in order, one at a time. Do not invent, replace, or reorder a problem. The data: a student guesses at random on a 10-question quiz with four options per question, so n = 10, p = 0.25, and X counts the correct answers; the binomial table gives 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 standard deviation is sqrt(10 x 0.25 x 0.75) = 1.37 (rounded), and about 4.33 for a 100-question version.
PROBLEM P1: 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?
PROBLEM P2: 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?
PROBLEM P3: 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?
PROBLEM P4: 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)?
After the four answers, I must briefly interpret each one: which part of the binomial model I used, what the value means for a guesser, and one use or limit of it; for P3, use the standard deviation of about 1.37 to say how far a typical score lands from the mean; for P4, check np and n(1 - p) against 10 for both the 10-question quiz and the 100-question quiz, say whether a normal curve may stand in for each, and state the 100-question curve as a model with its mean and standard deviation. I must then write a 3-5 sentence message to a tutoring-center coordinator who is new to statistics, explaining whether a score of 5 out of 10 is surprising for a pure guesser, giving a reason tied to the guessing numbers, and naming one limit. I must explain technical terms in everyday words.

SCORING CRITERIA: Score only evidence I supply: correct probability computations (40 points), model choice and probability reasoning (32 points), clarity for a non-expert reader (16 points), and showing work so it can be checked (12 points). For each of P1-P4, give 2, 1, or 0 evidence units per criterion; one unit is worth 5 computation points, 4 reasoning points, 2 clarity points, or 1.5 showing-work points. Sum all four problems within each criterion. Count my final message only with P4, not again with P1-P3.
COMPUTATION UNITS: A correct probability, mean, or count = 2; the right values and operation with only an arithmetic slip = 1; a missing result or the wrong values or operation = 0. Do not penalize an alternative valid method or equivalent correct notation, such as a percentage instead of a decimal.
REASONING UNITS: For P1-P3 separately, an accurate account of the binomial model or rule behind the value and what it means plus a use or limit tied to the guessing quiz = 2; exactly one of the two = 1; neither = 0. For P4, BOTH an accurate rule-of-thumb check of np and n(1 - p) for the 10-question and 100-question quizzes with the normal curve stated as a model AND a final message with a reason tied to the guessing numbers and a named limit = 2. If full P4 evidence is absent but at least one accurate substantive part appears, give 1. No accurate reasoning = 0; naming a rule alone does not earn a unit. Treating the mean 2.5 as one quiz's score, subtracting the running total at 5 for "at least 5", or checking the rule of thumb on n alone does not count as accurate. An arithmetic slip alone is not a reasoning error.
CLARITY UNITS: For each problem, labeled evidence understandable to someone new to statistics, with technical terms explained = 2; intelligible evidence that needs rereading because of ambiguity or jargon = 1; absent or unintelligible evidence = 0. Include the final message with P4 if submitted. Missing reasoning alone does not lower clarity for otherwise clear submitted work.
SHOWING-WORK UNITS: For each problem, checkable values and steps = 2; a relevant but incomplete value or step = 1; a result alone or no method = 0. A visible method stays checkable despite an arithmetic slip.

Do NOT invent grading rules.

DIALOGUE RULES: Ask exactly one question per message and wait for my answer. Start by asking for my complete first attempt. Build each next question from my exact words. Probe an unsupported claim and ask me to examine its evidence. Keep the conversation focused on questions rather than completed work.

ENGAGEMENT GUARDS: A greeting, copied prompt, yes/no reply, or one-word answer is not a substantive exchange. Do not flatter or accept agreement as evidence. Praise only a concrete move I made. If evidence is missing, say what is missing and ask for it. Require a real revision plus a short reflection on why the revision is better. Do not issue the report early.

EXIT CONDITION: Continue until at least five substantive exchanges have occurred AND I have supplied all four answers and their brief interpretations AND written my message to the coordinator AND revised at least one calculation or explanation AND reflected on the revision. Stop only after whichever requirement happens last.

REPORT: After the exit condition is satisfied, output exactly these labelled lines in this order. Replace X with the exact earned score; it may end in .5 and must not be rounded to a whole number. Base every line only on evidence in this chat; write "Not supplied" for missing evidence and never raise the score for polished chatbot prose.
STUDENT'S SCORE: X/100
EVIDENCE FOR THE SCORE: [specific evidence from my calculations and reasoning]
REVISION MADE: [what I changed and why]
STRENGTH: [one earned strength]
NEXT STEP: [one concrete improvement]
Then tell me to submit this report, my revised binomial report, and the share link.
COPY EVERYTHING ABOVE THIS LINE

Deliverables

Submit these three items in the assignment entry:

• your revised binomial report with all four answers, their brief interpretations, and your message to the coordinator — written, or recorded as audio or video 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;
• the coach's five-line report; and
• the conversation share link.

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.

Equivalent route (no AI required)

This route assesses the same Week 7 skills — finding a cumulative binomial probability, using the complement for an "at least" chance, computing and interpreting the binomial mean, and checking the large-counts rule of thumb before letting a normal curve stand in — for comparable time and effort, about 60 to 90 minutes, the same as the coached route. Complete the same four problems, the four brief interpretations, and the message to the coordinator. Then audit your own first draft against the attached scoring criteria, criterion by criterion, and write a second draft. Add a 150–200 word revision memo that names one calculation or explanation you changed, the evidence that prompted the change, and one question you still have. Submit the revised binomial report, your completed criteria checklist, and the revision memo in the assignment entry; together they replace the share link and the coach's report. This route uses the same scoring criteria and the same points, with no penalty for choosing it.

How you'll be graded

The full analytic rubric is attached to this assignment in Canvas.

Academic integrity and AI use

The calculations, reasoning, revision, and final report must be your own. Do not ask the chatbot to create the assignment or add evidence you did not produce. Submit an honest share link from your own dialogue. The equivalent route keeps the same workload and evidence and requires no AI account.

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