In an election there are two candidates. Being a statistician, you are interested in predicting the result of the election. So, you plan to conduct a survey. Using the learning skill of this course answer the following question. How many people should be surveyed to be at least 90% sure that the estimate is within 0.03 of the true value?

To determine how many people should be surveyed to be at least 90% sure that the estimate is within 0.03 of the true value in the context of an election between two candidates, we need to apply the concepts from probability theory and sampling distributions, especially the properties of the binomial distribution and the normal approximation to the binomial distribution.

We are dealing with a situation where each person surveyed can give one of two possible responses: either they support candidate A or they do not (i.e., they support candidate B). This naturally sets the stage for the binomial distribution, as we have a series of independent trials (each person surveyed), each with a binary outcome.

Understanding the Problem

Let the true proportion of voters who will vote for candidate A be denoted by p. Our goal is to estimate p with a sample proportion p^\hat{p}, obtained by surveying a certain number n of people.

We want to determine the sample size n such that we can be at least 90% confident that the estimate p^\hat{p} is within 0.03 of the true value of p. This is a classic example of a confidence interval problem for a population proportion.

Step-by-Step Mathematical Formulation

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