Week 13 Quiz

In a significance test, what does the null hypothesis H₀ state?

Select one.

• The claim the researcher hopes to find evidence for
• The claim of no effect that the test assumes is true
• The value of the statistic observed in the sample
• The probability that the data happened by chance

A student says they guessed at random on all 10 questions of a quiz with four options per question, and they scored 7. Let p be the student's true chance of answering a question correctly. Which pair of hypotheses tests the student's claim?

Select one.

• H₀: p = 0.25 and Hₐ: p > 0.25
• H₀: p > 0.25 and Hₐ: p = 0.25
• H₀: p̂ = 0.7 and Hₐ: p̂ > 0.25
• H₀: p = 0.7 and Hₐ: p < 0.7

A student says they guessed at random on all 10 questions of a quiz with four options per question, and they scored 7. If the student is guessing, the count X of correct answers is binomial with n = 10 and p = 0.25, and the table gives P(X = 7) = 0.0031, P(X = 8) = 0.0004, P(X = 9) = 0.0000, and P(X = 10) = 0.0000, each rounded to four decimals. What is the p-value P(X ≥ 7)? Enter the number rounded to four decimals.

Type a short answer.

A second student who also claims to have guessed scores 3 of 10 on the same quiz. Under the guessing model (n = 10, p = 0.25), P(X ≤ 2) = 0.5256 and P(X = 3) = 0.2503. What is the p-value P(X ≥ 3), rounded to two decimals?

Select one.

• 0.53
• 0.25
• 0.78
• 0.47

A student says they guessed at random on all 10 questions of a quiz with four options per question, and they scored 7. The p-value is P(X ≥ 7) = 0.0035 under H₀: p = 0.25. Which sentence interprets the p-value correctly?

Select one.

• There is a 0.35% chance that the student was purely guessing on the quiz.
• There is a 0.35% chance that the score of 7 happened by luck alone.
• If the student were purely guessing, a score of 7 or more would occur on about 0.35% of quizzes.
• The student's edge over guessing is large, because 0.0035 is a very small number.

A different student who claims to have guessed scores 5 of 10. The p-value is P(X ≥ 5) = 0.0781 under H₀: p = 0.25, and the significance level chosen in advance is α = 0.05. Which decision and conclusion follow?

Select one.

• Reject H₀; the score shows that the student was not guessing.
• Fail to reject H₀; the score is consistent with guessing.
• Accept H₀; the score proves that the student was guessing.
• Reject H₀, because 0.0781 is below a looser α of 0.10.

True or false: a student scores 5 of 10, the p-value is 0.0781, and the test fails to reject H₀: p = 0.25 at α = 0.05. This shows that the student was purely guessing.

True or false.

• True
• False

A teacher flags any student who claims to have guessed when the score is high. Rule A flags a score of 5 or more; Rule B flags a score of 7 or more. For a pure guesser, P(X ≥ 5) = 0.0781 and P(X ≥ 7) = 0.0035. Which statement compares the two rules correctly?

Select one.

• Rule A has the lower Type I error rate, because it flags more students.
• Rule B has the lower Type I error rate and also makes fewer Type II errors.
• Both rules have a Type I error rate of 0.05, because α is always 0.05.
• Rule B has the lower Type I error rate but makes more Type II errors.

One student scores 4 of 10 (40%) on the guessing quiz; the p-value is 0.2241. Another scores 40 of 100 (40%) on a 100-question version, where guessing has mean 25 and standard deviation 4.3301; the score sits about 3.46 standard deviations above 25, and the p-value is well below 0.01. Which conclusion is best?

Select one.

• The longer quiz is strong evidence of an edge, yet the edge is still a modest 40% versus 25%.
• The longer quiz shows a much bigger edge over guessing, because its p-value is much smaller.
• The short quiz proves that its student was guessing, because 0.2241 is above 0.05.
• The two results disagree, so one of the two p-values must have been computed wrongly.

A student says they guessed at random on all 10 questions of a quiz with four options per question, and they scored 7. The p-value is P(X ≥ 7) = 0.0035 under H₀: p = 0.25. Select every statement that is true.

Select every option that applies.

• It was computed by assuming that H₀: p = 0.25 is true.
• It is the probability that the student was purely guessing.
• It is below α = 0.01, so the test rejects H₀ at that level too.
• It proves that the student studied the material for the quiz.
• It measures how much more the student knows than a guesser.