Margin of Error for 60% Result

This question is about interpreting a polling result that reports a percentage in favor of a candidate based on a finite sample. The setup involves a simple opinion poll with a clear split between support for one option and support for all others, and asks how to quantify the uncertainty around the reported percentage. It is the kind of question used in statistics screenings, data-focused interviews, and quant research roles where candidates are expected to translate a verbal description of a survey into a clean probabilistic model and express how reliable the headline number is.

Answering it relies on understanding proportions as averages of Bernoulli trials, the binomial distribution, and the normal approximation to that distribution for large samples. It leans on the notion of standard error, the construction of confidence intervals, and the use of an appropriate critical value from the normal distribution. Interviewers watch for correct identification of the relevant proportion, careful handling of sample size, an ability to justify the approximation being used, and a clear interpretation of what "margin of error" actually means in probabilistic terms.

What it tests

Whenever estimating a population proportion from a random sample, the sampling distribution of the sample proportion is approximately normal (by the Central Limit Theorem) when the sample size is large, with mean equal to the true proportion and standard deviation (standard error) given by $\sqrt{p(1-p)/n}$. The margin of error at a given confidence level (commonly 95%) is found by multiplying this standard error by the appropriate z-score (about 2 for 95%). This structure holds because each response is an independent Bernoulli trial, so the sum is binomial, and the binomial's variance naturally leads to this formula. The normal approximation becomes accurate as $n$ increases, making this method robust for large surveys.