Questions: 8. Probability and Statistics - Decision Theory An instructor gives a short quiz involving 10 true-false questions. To test the hypothesis that students are guessing, she adopts the following decision rule: If seven or more answers are correct, the student is not guessing, else the student is guessing. Find the probability of rejecting the hypothesis, when it is correct. Pick ONE option 0.08 0.17 0.26 0.35 Clear Selection

Solution Steps

To solve this problem, we need to calculate the probability of a student getting 7 or more correct answers by guessing. Since each question is true-false, the probability of guessing correctly is 0.5. We can model this situation using a binomial distribution with parameters \( n = 10 \) (number of questions) and \( p = 0.5 \) (probability of guessing correctly). The probability of rejecting the hypothesis when it is correct is the sum of probabilities of getting 7, 8, 9, or 10 correct answers.

We are given a quiz with 10 true-false questions. We need to find the probability of rejecting the hypothesis that students are guessing, given that the hypothesis is correct. The decision rule is to reject the hypothesis if a student answers 7 or more questions correctly.

The situation can be modeled using a binomial distribution with parameters \( n = 10 \) (number of questions) and \( p = 0.5 \) (probability of guessing correctly). We are interested in the probability of getting 7 or more correct answers.

The probability of getting exactly \( k \) correct answers is given by the binomial probability mass function: \[ P(X = k) = \binom{n}{k} p^k (1-p)^{n-k} \] We need to calculate the sum of probabilities for \( k = 7, 8, 9, \) and \( 10 \): \[ P(X \geq 7) = P(X = 7) + P(X = 8) + P(X = 9) + P(X = 10) \]

Using the binomial distribution, the probability of rejecting the hypothesis when it is correct is: \[ P(X \geq 7) = 0.1719 \]

Final Answer