During the Last Few Days Before a Presidential Election, There $Y$

Essay

During the last few days before a presidential election, there is a frenzy of voting
intention surveys.On a given day, quite often there are conflicting results from three
major polls.
(a) Think of each of these polls as reporting the fraction of successes (1s) of a Bernoulli random variable $Y$ , where the probability of success is $\operatorname { Pr } ( Y = 1 ) = p$ . Let $\hat { p }$ be the fraction of successes in the sample and assume that this estimator is normally distributed with a mean of $p$ and a variance of $\frac { p ( 1 - p ) } { n }$ . Why are the results for all polls different, even though they are taken on the same day?