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 * Pn? 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 pN 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?
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 * Pn? 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 pN 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?
Calculus For The Life Sciences
2nd Edition
ISBN:9780321964038
Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.
Chapter12: Probability
Section12.2: Introduction To Probability
Problem 52E
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