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Suppose you want to find out how many people support Policy X. A standard polling approach is to just ask N many people whether or not they support Policy X, and take the fraction of people who say yes as an estimate of the probability that any one person supports the policy. Suppose that the probability someone supports the policy is p, which you do not know. Let py be the number of people polled who supported the policy, divided by the total number of people polled N. 1) What distribution of N * PN? 2) Show that the expected value of pN is p. 3) If I want my estimate to be accurate, I want the error of pN to be small. How many people should I poll to guarantee the expected squared error on py is less than e? 4) How many people should I poll to guarantee the expected squared error on py is less than e, if I don't know p? 5) Just because the expected error is small doesn't mean the actual error is small. How many people should I poll to guarantee that the actual error on py is less than with 90% confidence? 6) How many people should I poll to guarantee the actual error on py is less than with 90% confidence, if I don't know p?

Suppose you want to find out how many people support Policy X. A standard polling approach is to just ask N many people whether or not they support Policy X, and take the fraction of people who say yes as an estimate of the probability that any one person supports the policy. Suppose that the probability someone supports the policy is p, which you do not know. Let py be the number of people polled who supported the policy, divided by the total number of people polled N. 1) What distribution of N * PN? 2) Show that the expected value of pN is p. 3) If I want my estimate to be accurate, I want the error of pN to be small. How many people should I poll to guarantee the expected squared error on py is less than e? 4) How many people should I poll to guarantee the expected squared error on py is less than e, if I don't know p? 5) Just because the expected error is small doesn't mean the actual error is small. How many people should I poll to guarantee that the actual error on py is less than with 90% confidence? 6) How many people should I poll to guarantee the actual error on py is less than with 90% confidence, if I don't know 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
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Suppose you want to find out how many people support Policy X. A standard polling approach is to just ask N many people whether or not they support Policy X, and take the fraction of people who say yes as an estimate of the probability that any one person supports the policy. Suppose that the probability someone supports the policy is p, which you do not know. Let py be the mumber of people polled who supported the policy, divided by the total number of people polled N. 1) What distribution of N * PN? 2) Show that the expected value of pN is p. 3) If I want my estimate to be accurate, I want the error of pn to be small. How many people should I poll to guarantee the erpected squared error on py is less than e? 4) How many people should I poll to guarantee the expected squared error on pn is less than e, if I don't know p? 5) Just because the expected error is small doesn't mean the actual error is small. How many people should I poll to guarantee that the actual error on pN is less than e with 90% confidence? 6) How many people should I poll to guarantee the actual error on pN is less than e with 90% confidence, if I don't know p?
Transcribed Image Text:Suppose you want to find out how many people support Policy X. A standard polling approach is to just ask N many people whether or not they support Policy X, and take the fraction of people who say yes as an estimate of the probability that any one person supports the policy. Suppose that the probability someone supports the policy is p, which you do not know. Let py be the mumber of people polled who supported the policy, divided by the total number of people polled N. 1) What distribution of N * PN? 2) Show that the expected value of pN is p. 3) If I want my estimate to be accurate, I want the error of pn to be small. How many people should I poll to guarantee the erpected squared error on py is less than e? 4) How many people should I poll to guarantee the expected squared error on pn is less than e, if I don't know p? 5) Just because the expected error is small doesn't mean the actual error is small. How many people should I poll to guarantee that the actual error on pN is less than e with 90% confidence? 6) How many people should I poll to guarantee the actual error on pN is less than e with 90% confidence, if I don't know p?
A potential solution to this is the following: suppose that 'YES' is the embarrassing or socially shameful answer; give the people you are polling the following instructions: flip a coin privately, and if it comes up heads answer honestly, but if it comes up tails answer 'YES' regardless of what the truth is. This gives people plausible deniability about why they answered yes, if pressed. Again, let py be the fraction of people who said 'YES'. Let p be the probability that a randomly selected person says 'yes'; let q be the probability that a person's true answer is 'yes'. Note again, we have that the expected value of pN is p, but we are actually interested in measuring q. 7) What is the relationship between q and p? 8) Construct an estimator ĝy from pN so that the expected value of ĝy is q. 9) If I want my estimate to be accurate, I want the error on ĝn to be small. How many people should I poll to guarantee that the expected squared error on qn is less than e, when I don't know the value of q? How does this compare to the answer in the previous section? 10) How many people should I poll to guarantee the actual error on ĝn is less than e, with 90% confidence, if I don't know q? 11) What is the additional 'cost' of accurate polling if I want to preserve people's privacy in this way?
Transcribed Image Text:A potential solution to this is the following: suppose that 'YES' is the embarrassing or socially shameful answer; give the people you are polling the following instructions: flip a coin privately, and if it comes up heads answer honestly, but if it comes up tails answer 'YES' regardless of what the truth is. This gives people plausible deniability about why they answered yes, if pressed. Again, let py be the fraction of people who said 'YES'. Let p be the probability that a randomly selected person says 'yes'; let q be the probability that a person's true answer is 'yes'. Note again, we have that the expected value of pN is p, but we are actually interested in measuring q. 7) What is the relationship between q and p? 8) Construct an estimator ĝy from pN so that the expected value of ĝy is q. 9) If I want my estimate to be accurate, I want the error on ĝn to be small. How many people should I poll to guarantee that the expected squared error on qn is less than e, when I don't know the value of q? How does this compare to the answer in the previous section? 10) How many people should I poll to guarantee the actual error on ĝn is less than e, with 90% confidence, if I don't know q? 11) What is the additional 'cost' of accurate polling if I want to preserve people's privacy in this way?
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