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3. Drink spiking means to put alcohol or drugs into someone's drink without their knowledge or permission. Suppose we want to estimate p, the population proportion of University of South Carolina undergraduates who have participated in this activity. We regard p as a population- level parameter which is unknown (0

3. Drink spiking means to put alcohol or drugs into someone's drink without their knowledge or permission. Suppose we want to estimate p, the population proportion of University of South Carolina undergraduates who have participated in this activity. We regard p as a population- level parameter which is unknown (0

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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3. Drink spiking means to put alcohol or drugs into someone's drink without their knowledge or permission. Suppose we want to estimate p, the population proportion of University of South Carolina undergraduates who have participated in this activity. We regard p as a population- level parameter which is unknown (0 < p < 1). One strategy to estimate p is to sample n undergraduates and ask each one the following question directly: Have you ever drink-spiked someone? Let Y₁, Y2,..., Yn denote the "1/0" ("yes/no") responses and assume Y₁, Y2, ..., Yn is an iid sample from a Bernoulli population with mean p; i.e., the population probability mass function (pmf) is PY (y) = - {2²²- p³ (1-p)¹, y = 0,1 0, otherwise. Note that under this model, py (1) = p = P("yes") and py (0) = 1 - p = P("no"). (a) Show the log-likelihood function is n n In L(ply) = y; Inp+ (n - Σy: 1n(1 – p). i=1 i=1 Maximize In L(ply) to show the MLE of p is n Y = ² Yi, n the proportion of "1's" in the sample; i.e., the sample proportion.
Transcribed Image Text:3. Drink spiking means to put alcohol or drugs into someone's drink without their knowledge or permission. Suppose we want to estimate p, the population proportion of University of South Carolina undergraduates who have participated in this activity. We regard p as a population- level parameter which is unknown (0 < p < 1). One strategy to estimate p is to sample n undergraduates and ask each one the following question directly: Have you ever drink-spiked someone? Let Y₁, Y2,..., Yn denote the "1/0" ("yes/no") responses and assume Y₁, Y2, ..., Yn is an iid sample from a Bernoulli population with mean p; i.e., the population probability mass function (pmf) is PY (y) = - {2²²- p³ (1-p)¹, y = 0,1 0, otherwise. Note that under this model, py (1) = p = P("yes") and py (0) = 1 - p = P("no"). (a) Show the log-likelihood function is n n In L(ply) = y; Inp+ (n - Σy: 1n(1 – p). i=1 i=1 Maximize In L(ply) to show the MLE of p is n Y = ² Yi, n the proportion of "1's" in the sample; i.e., the sample proportion.
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 drink-spiking. 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 drink-spiked someone?" Students who flip T will answer "yes/no" to "Is the last digit of your SSN 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 0 = 0.5P+0.5(0.3) = 0.5p+ 0.15. The novelty of this strategy is that a "yes" response from any student is not necessarily incrim- inating. S/he could be answering "yes" to the second question! (b) Explain where the expression for above comes from. Hint: Recall the Law of Total Probability from Chapter 2 and assume all final SSN digits are equally likely. Describe how you would estimate p using maximum likelihood under this second strategy. (c) Let pi 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 to determine this and graph the relative efficiency as a function of p.
Transcribed Image Text: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 drink-spiking. 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 drink-spiked someone?" Students who flip T will answer "yes/no" to "Is the last digit of your SSN 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 0 = 0.5P+0.5(0.3) = 0.5p+ 0.15. The novelty of this strategy is that a "yes" response from any student is not necessarily incrim- inating. S/he could be answering "yes" to the second question! (b) Explain where the expression for above comes from. Hint: Recall the Law of Total Probability from Chapter 2 and assume all final SSN digits are equally likely. Describe how you would estimate p using maximum likelihood under this second strategy. (c) Let pi 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 to determine this and graph the relative efficiency as a function of p.
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