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Upholding academic integrity in online classes is a daunting challenge of faculty members of the Visayas State University. Academic dishonesty undermines assessments along with the entire learning system. It creates the illusion that the learner has mastered the subject matter when that is not the case (https://collegiseducation.com/news/online-learning/academic-integrity-in-remote-learning/). It is a common belief that a considerable proportion of students cheat. Suppose we want to estimate p, the proportion of VSU undergraduate students who cheat. One strategy to estimate p is to sample n undergraduate students and ask each one the following question directly: Have you ever cheated in answering one of your class requirements (quizzes, exams, assignments, projects, etc.? Let Y₁, Y₂,, Yn denote the "1/0" ("yes/no") responses. a. What is the distribution of Y₁, i=1,2,...,n? b. Based on your answer in (a), find the maximum likelihood estimator of p.

Upholding academic integrity in online classes is a daunting challenge of faculty members of the Visayas State University. Academic dishonesty undermines assessments along with the entire learning system. It creates the illusion that the learner has mastered the subject matter when that is not the case (https://collegiseducation.com/news/online-learning/academic-integrity-in-remote-learning/). It is a common belief that a considerable proportion of students cheat. Suppose we want to estimate p, the proportion of VSU undergraduate students who cheat. One strategy to estimate p is to sample n undergraduate students and ask each one the following question directly: Have you ever cheated in answering one of your class requirements (quizzes, exams, assignments, projects, etc.? Let Y₁, Y₂,, Yn denote the "1/0" ("yes/no") responses. a. What is the distribution of Y₁, i=1,2,...,n? b. Based on your answer in (a), find the maximum likelihood estimator of p.

MATLAB: An Introduction with Applications
MATLAB: An Introduction with Applications
6th Edition
ISBN:9781119256830
Author:Amos Gilat
Publisher:Amos Gilat
Chapter1: Starting With Matlab
Section: Chapter Questions
Problem 1P
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Upholding academic integrity in online classes is a daunting challenge of faculty members of the Visayas State University. Academic dishonesty undermines assessments along with the entire learning system. It creates the illusion that the learner has mastered the subject matter when that is not the case (https://collegiseducation.com/news/online-learning/academic-integrity-in-remote-learning/). It is a common belief that a considerable proportion of students cheat. Suppose we want to estimate p, the proportion of VSU undergraduate students who cheat. One strategy to estimate p is to sample n undergraduate students and ask each one the following question directly: Have you ever cheated in answering one of your class requirements (quizzes, exams, assignments, projects, etc.? Let Y₁, Y₂,..., Yn denote the "1/0" ("yes/no") responses. a. What is the distribution of Y₁, i = 1,2,...,n? b. Based on your answer in (a), find the maximum likelihood estimator of p. c. An obvious problem with the strategy above is that students may not be truthful in their responses; e.g., a student may respond "no" when, in fact, s/he is guilty of cheating. As an alternative strategy, suppose we ask students to first flip an unbiased coin (H/T) and have them hide the result from us (i.e., do not reveal to us the outcome of the coin). • Students who flip H will answer "yes/no" to "Have you ever cheated in answering one of your class requirements (quizzes, exams, assignments, projects, etc.?" • Students who flip T will answer "yes/no" to "Is the last digit of your ID number a 0, 1, or 2?" Let X₁, X2,,Xn denote the "1/0" ("yes/no") responses from this strategy and assume that X₁, X2,... Xn is an iid sample from a Bernoulli population with mean 7 0 = 0.5p+ 0.15. The novelty of this strategy is that a "yes" response from any student is not necessarily incriminating. S/he could be answering "yes" to the second question! Describe how you would estimate p using maximum likelihood method under this second strategy. d. Let p₁ denote the MLE of p from the first strategy. Let p2 denote the MLE of p from the second strategy. If all students' responses are truthful under both strategies (i.e., no one lies), then which estimator is more efficient? Calculate the relative efficiency as a function of p.
Transcribed Image Text:Upholding academic integrity in online classes is a daunting challenge of faculty members of the Visayas State University. Academic dishonesty undermines assessments along with the entire learning system. It creates the illusion that the learner has mastered the subject matter when that is not the case (https://collegiseducation.com/news/online-learning/academic-integrity-in-remote-learning/). It is a common belief that a considerable proportion of students cheat. Suppose we want to estimate p, the proportion of VSU undergraduate students who cheat. One strategy to estimate p is to sample n undergraduate students and ask each one the following question directly: Have you ever cheated in answering one of your class requirements (quizzes, exams, assignments, projects, etc.? Let Y₁, Y₂,..., Yn denote the "1/0" ("yes/no") responses. a. What is the distribution of Y₁, i = 1,2,...,n? b. Based on your answer in (a), find the maximum likelihood estimator of p. c. An obvious problem with the strategy above is that students may not be truthful in their responses; e.g., a student may respond "no" when, in fact, s/he is guilty of cheating. As an alternative strategy, suppose we ask students to first flip an unbiased coin (H/T) and have them hide the result from us (i.e., do not reveal to us the outcome of the coin). • Students who flip H will answer "yes/no" to "Have you ever cheated in answering one of your class requirements (quizzes, exams, assignments, projects, etc.?" • Students who flip T will answer "yes/no" to "Is the last digit of your ID number a 0, 1, or 2?" Let X₁, X2,,Xn denote the "1/0" ("yes/no") responses from this strategy and assume that X₁, X2,... Xn is an iid sample from a Bernoulli population with mean 7 0 = 0.5p+ 0.15. The novelty of this strategy is that a "yes" response from any student is not necessarily incriminating. S/he could be answering "yes" to the second question! Describe how you would estimate p using maximum likelihood method under this second strategy. d. Let p₁ denote the MLE of p from the first strategy. Let p2 denote the MLE of p from the second strategy. If all students' responses are truthful under both strategies (i.e., no one lies), then which estimator is more efficient? Calculate the relative efficiency as a function of p.
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