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: 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 hoestly, 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 în be the fraction of people who said 'YES'. Let p be the probability that a randomly selected person says ʻyes'; let q be the probability that a person's true answer is ʻyes’. Note again, we have that the expected value of pn is p, but we are actually interested in measuring q.

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What is the relationship between q and 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 \( \hat{p}_N \) be the number of people polled who supported the policy, divided by the total number of people polled \( N \).
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 \( \hat{p}_N \) be the number of people polled who supported the policy, divided by the total number of people polled \( N \).
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: 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 \(\hat{p}_N\) 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 \(\hat{p}_N\) is \(p\), but we are actually interested in measuring \(q\).
Transcribed Image Text: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: 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 \(\hat{p}_N\) 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 \(\hat{p}_N\) is \(p\), but we are actually interested in measuring \(q\).
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