# Inference from sample proportion to population proportion

I hope I'm posting in the right forum! My daughter has something like the following problem in her college math class: In a random sample of 100 registered voters, 75 people say they will vote for Clinton (the assignment was made up before the election). Using statistical inference, what is the probability that Clinton will receive more than 60 percent of votes in the general election?

I have changed the numbers out of general principle. I hope that it doesn't affect the problem materially. I don't see how to get the answer to this kind of problem. I understand confidence intervals, but I don't see how to get from that to an answer. I hope that I correctly classified the type of problem in the title to the post.

Thanks,

Mark

## 1 Answer

As the question is worded, it's hard to see a frequentist solution, because a frequentist would presumably treat the proportion who are voting for Clinton as a fixed, single value of the population. But it can be interpreted in a Bayesian fashion like this: if you get 75 successes from 100 independent Bernoulli trials with a true probability of success $θ$, then what is the posterior probability that $θ > .6$? Use a conjugate prior for $θ$ (which will be a beta distribution) and you can get a closed form for the posterior distribution of $θ$, from which you can take the .6 quantile to answer the question.