Problem 1: Asking Embarrassing Questions, Politely When doing polling, for instance to figure out how popular a given candidate is, a common trick is to just ask N many people whether they support that candidate, and take the support to be the faction of people who say yes: if 70 people support the candidate out of 100 asked, we estimate the support at 70% or 0.7. Suppose that the probability a person supports a candidate is P, which you do not know. Let pN be the fraction of N people polled who support the candidate: total supporters divided by N people polled. 1) What is the distribution of N * p,? 2) Show hat the expected value of PN is p, i.e., PN is a valid estimator for p.
Problem 1: Asking Embarrassing Questions, Politely When doing polling, for instance to figure out how popular a given candidate is, a common trick is to just ask N many people whether they support that candidate, and take the support to be the faction of people who say yes: if 70 people support the candidate out of 100 asked, we estimate the support at 70% or 0.7. Suppose that the probability a person supports a candidate is P, which you do not know. Let pN be the fraction of N people polled who support the candidate: total supporters divided by N people polled. 1) What is the distribution of N * p,? 2) Show hat the expected value of PN is p, i.e., PN is a valid estimator for p.
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![Problem 1: Asking Embarrassing Questions, Politely
When doing polling, for instance to figure out how popular a given candidate is, a common trick is to just ask N
many people whether they support that candidate, and take the support to be the faction of people who say yes: if 70
people support the candidate out of 100 asked, we estimate the support at 70% or 0.7. Suppose that the probability
a person supports a candidate is p, which you do not know. Let PN be the fraction of N people polled who support
the candidate: total supporters divided by N people polled.
1) What is the distribution of N * p,?
2) Show hat the expected value of pN is p, i.e., pN is a valid estimator for p.
If you want your estimated value of p to be accurate, you want your 'error' on pN to be small.
3) How many people N should you poll to guarantee the erpected squared error on pN is less than e?
4) How many people N should you poll to guarantee the expected squared error on py is less than e, even if you
don't know p?
In the previous to problems, we considered the average or expected squared error. But just because the expected
error is small doesn't mean the actual error is small.
5) How many people N should you poll to guarantee the actual error on PN is less than e with 95% confidence,
even if you don't know p?
However, a problem with polling is whether or not people are willing to answer honestly. If a question might be
viewed as shameful or embarrassing (about politics, sexual activity, or whatever people are sensitive about), they
may be reluctant to answer honestly.
A potential solution to this is the following: when asking whether they support or do not support a given candidate,
give the people you are polling the following instructions: flip a fair coin privately, and if it comes up HEADS, answer
honestly; if it comes up TAILS, flip another fair coin and if it comes up HEADS, answer 'support', if it comes up
TAILS, answer do not 'support'. In this case, the person being polled can always claim that whatever they answered
was the result of the coin in a sense, the results are anonymized and the people being polled are protected.
Let p be the probability that a randomly polled person using this method says 'support'; let q be the true probability
a random person actually supports the candidate. We would like to know the value of q, but we can only estimate
the value of p: let p, be the fraction of N people who answer 'support' using this method. We have that ElPN] = p,
as before.
7) What is the relationship between q and p?
8) Construct an estimator qy from PN (i.e., a formula for ĝn in terms of pN) so that the expected value ElâN] = q.
What is the variance of ĝn?
9) How many people N should you poll to guarantee the actual error between ĝn and q is less than e, with 90%
confidence? Note, q is unknown, so you cannot use it to determine N.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1d48ea68-6d7d-476d-9ed5-87a181f9a485%2Fa1d1270e-a14e-4be2-823d-01d0e20c2905%2Ffkb6t6_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Problem 1: Asking Embarrassing Questions, Politely
When doing polling, for instance to figure out how popular a given candidate is, a common trick is to just ask N
many people whether they support that candidate, and take the support to be the faction of people who say yes: if 70
people support the candidate out of 100 asked, we estimate the support at 70% or 0.7. Suppose that the probability
a person supports a candidate is p, which you do not know. Let PN be the fraction of N people polled who support
the candidate: total supporters divided by N people polled.
1) What is the distribution of N * p,?
2) Show hat the expected value of pN is p, i.e., pN is a valid estimator for p.
If you want your estimated value of p to be accurate, you want your 'error' on pN to be small.
3) How many people N should you poll to guarantee the erpected squared error on pN is less than e?
4) How many people N should you poll to guarantee the expected squared error on py is less than e, even if you
don't know p?
In the previous to problems, we considered the average or expected squared error. But just because the expected
error is small doesn't mean the actual error is small.
5) How many people N should you poll to guarantee the actual error on PN is less than e with 95% confidence,
even if you don't know p?
However, a problem with polling is whether or not people are willing to answer honestly. If a question might be
viewed as shameful or embarrassing (about politics, sexual activity, or whatever people are sensitive about), they
may be reluctant to answer honestly.
A potential solution to this is the following: when asking whether they support or do not support a given candidate,
give the people you are polling the following instructions: flip a fair coin privately, and if it comes up HEADS, answer
honestly; if it comes up TAILS, flip another fair coin and if it comes up HEADS, answer 'support', if it comes up
TAILS, answer do not 'support'. In this case, the person being polled can always claim that whatever they answered
was the result of the coin in a sense, the results are anonymized and the people being polled are protected.
Let p be the probability that a randomly polled person using this method says 'support'; let q be the true probability
a random person actually supports the candidate. We would like to know the value of q, but we can only estimate
the value of p: let p, be the fraction of N people who answer 'support' using this method. We have that ElPN] = p,
as before.
7) What is the relationship between q and p?
8) Construct an estimator qy from PN (i.e., a formula for ĝn in terms of pN) so that the expected value ElâN] = q.
What is the variance of ĝn?
9) How many people N should you poll to guarantee the actual error between ĝn and q is less than e, with 90%
confidence? Note, q is unknown, so you cannot use it to determine N.
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