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Over half of the people

A government official wants to find out how many people will vote for each party in a coming election. She asks a sample of the voters. Suppose that 46% of the people will vote for a certain party. [For this question, assume that all voters sampled are independent, and use any suitable approximations for the distribution]

(a) Suppose that she asks 500 people. What is the probability that more than half of them say they will vote for this party.

(b) What is the smallest number n such that there is a more than 95% chance that the number of people sampled who say they will vote for this party is less than n.

(c) How many people should she sample to ensure that there is a less than 0.5% chance that over half of the people sampled say they will vote for this party?

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