Week 13 — Was the Student Guessing? A Short Significance-Test Report — Traditional alternative (no AI)

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

Assignment 12 of the term — find p-values from the guessing-quiz table, decide at a significance level chosen in advance, describe the two ways a test can be wrong, separate statistical from practical significance, and correct three misreadings of a p-value.

This week you learned to test a claim: to state a null and an alternative hypothesis, to find a p-value by assuming the null hypothesis is true, to decide at a significance level chosen in advance, to name the two ways a test can be wrong, and to tell statistical from practical significance. In Week 7 you found binomial chances for a student guessing on a 10-question quiz; this week the same table becomes a test of a claim. In this assignment you work four short problems, interpret each one, correct three flawed claims, and explain a result 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

Several students each say they guessed at random on every question of a quiz with four options per question. If a student is guessing, then p, the student's true chance of a correct answer on each question, is 1 ⁄ 4 = 0.25. On a 10-question quiz the count of correct answers X is then binomial with n = 10 and p = 0.25. Its null mean is np = 10 × 0.25 = 2.5 correct answers, and its standard deviation is about 1.37 correct answers. The guessing-quiz table (Dataset D) gives P(X = k) for k = 0, 1, 2, …, 10 correct answers, each rounded to four decimals: 0.0563, 0.1877, 0.2816, 0.2503, 0.1460, 0.0584, 0.0162, 0.0031, 0.0004, 0.0000, 0.0000. For every test below, use the significance level α = 0.05, chosen before anyone looked at a quiz.

Work these four problems, showing your arithmetic:

1. A student who says they guessed on all 10 questions of a four-option quiz scored 7. In the guessing-quiz table, P(X = 7) = 0.0031, P(X = 8) = 0.0004, and P(X = 9) and P(X = 10) both round to 0.0000. What is the p-value P(X ≥ 7), to four decimals?
2. A second student who says they guessed scored 5 on the same quiz. In the guessing-quiz table, P(X = 5) = 0.0584, P(X = 6) = 0.0162, P(X = 7) = 0.0031, P(X = 8) = 0.0004, and P(X = 9) and P(X = 10) both round to 0.0000. What is the p-value P(X ≥ 5), to four decimals?
3. A third student who says they guessed got 4 of the 10 questions right. In the guessing-quiz table, P(X ≤ 2) = 0.5256 and P(X = 3) = 0.2503. What is the p-value P(X ≥ 4) = 1 − (P(X ≤ 2) + P(X = 3)), to four decimals?
4. On a 100-question version of the four-option quiz, a pure guesser's count of correct answers has null mean 25 and standard deviation 4.3301. What count sits 3 standard deviations above the null mean, 25 + 3 × 4.3301?

Beside each answer, add one brief interpretation:

• Problem 1. State H₀ and Hₐ in symbols and in words about p, and say why the alternative is one-sided. Give your decision at α = 0.05 and a conclusion in context. Write the course's p-value sentence: "if the student were guessing, a score this high would occur about … of the time". Then correct Claim 1 below.
• Problem 2. Give your decision at α = 0.05 and say what it does and does not show about the second student, correcting Claim 2 below. Then take the rule "reject H₀ when the quiz has 5 or more correct answers". Describe a Type I error and a Type II error for that rule in the quiz's own terms, say which error your problem 2 value is the probability of, and explain what happens to each error if the cutoff rises to 7.
• Problem 3. Give your decision at α = 0.05, write the course's p-value sentence for this student, and name one limit of a 10-question quiz for telling a small edge apart from pure guessing.
• Problem 4. A fourth student gets 40 of the 100 questions right. Both np = 25 and n(1 − p) = 75 are at least 10, so a normal model for the count is reasonable. The empirical rule says about 99.7% of a normal model lies within 3 standard deviations of its mean. Compare 40 with the count you found, and use the empirical rule to say roughly how rare 40 or more correct answers would be for a guesser. Then explain why 4 right out of 10 and 40 right out of 100, the same proportion of correct answers, lead to such different p-values, and say whether that proportion is a large edge over guessing. Correct Claim 3 below.

Check a draft conclusion

A study partner drafted a summary of these results. It sounds confident, but each of its three claims misreads a significance test. Quote each claim in your report, say what is wrong with it, and rewrite it so it is correct.

• Claim 1: "The p-value for the score of 7 is the probability that the student was guessing. It is tiny, so the student almost certainly studied."
• Claim 2: "The second student's p-value is above 0.05, so the test proves that student was guessing."
• Claim 3: "The fourth student's p-value is far smaller than the third student's, so the fourth student knows far more of the material."

The American Statistical Association warns that a p-value is not the probability that the null hypothesis is true and does not measure the size or importance of an effect. Use that warning, and your own numbers, as evidence.

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 at a campus tutoring center has never studied statistics. After hearing about the fourth student, the volunteer says, "A p-value that small means this student really knows the material, right?" Write a 3–5 sentence message that answers the volunteer in everyday words, gives a reason tied to the quiz numbers, and names one limit — for example, that a p-value does not measure how much a student knows, or that a test cannot show why a student did well. Explain any technical term, such as "p-value" or "statistically significant", in everyday words. More than one good answer is possible.

For a different, ungraded example, imagine a game-night host who reads that a test of a die suspected of being loaded, with H₀: P(six) = 1 ⁄ 6, found no significant evidence against a fair die, and who asks whether the die is fair. Model answer: "Not necessarily. The test checked whether the rolls were too unusual for a fair die, and they were not, so the rolls are consistent with a fair die. A die that is only a little off could give rolls like these too, so the test cannot prove the die is fair. Rolling it many more times would give a sharper answer."

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

Deliverables

Submit a single significance-test report containing:

• the four answers, each with the arithmetic shown and a brief interpretation as the task describes;
• your corrections of the three claims, each quoting the claim, saying what is wrong with it, and rewriting it so it is correct;
• your message to the volunteer, with a reason tied to the quiz 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 to two pages. 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 guessing-quiz distribution with each tail you used shaded and 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 p-values, the decisions, the interpretations, the corrections, 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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