# Confidence interval and sample size multinomial probabilities

I'm an absolute beginner in statistics. Please excuse any wrong assumptions or missing information in my question. I have a question that relates to a multinomial distribution (not even 100% sure about this) that I hope somebody can help me with. If I take a sample (lets assume $n=400$) on a categorical variable that has more than two possible outcomes (e.g. blue, black, green, yellow) and plot the frequencies so that I can derive the probabilities. E.g.: black 10% blue 25% green 35% yellow 30%

How could I compute the 95% confidence interval for those probabilities? And how could I determine the sample size required in order to get an accurate result within +-3% for each probability? Please let me know if the answer to the question requires any additional information.

• Welcome to the website, you may want to do a search on maximum likelihood estimation and standard error, this link may be a good start. P.S: Although they are talking about a different distribution (Pareto) in the link, the concepts apply to your case. – Zhubarb Aug 10 '14 at 14:00
• Also check this out: sites.stat.psu.edu/~sesa/stat504/Lecture/lec3_4up.pdf – Zhubarb Aug 10 '14 at 14:05
• Would you know how to do it if you got only two categories instead of four? – Michael M Aug 10 '14 at 14:12
• Hi Michael, I think in this case it could work with a binomial distribution and I would use the normal distribution (since it's approximately the same) to calculate the confidence interval. Please correct me if I'm wrong. – Dirk Aug 10 '14 at 16:06
• Then you can simply do this for each category separately (e.g. black vs. non-black). – Michael M Aug 10 '14 at 17:04