# 95% confidence interval for the proportion and population size

A simple random sample responses from n = 271 residents of a province. Of these, 57 report being current members of a club.

Give a 95% confidence interval for the proportion of residents who are club members. Also there are totally 800,000 club members, obtain a 95% confidence interval for the population size of the province.

My answer (16.181, 25.885) for the first part is incorrect, is it because my answer shouldn't be in percentage? Any thoughts on the second part?

Thanks!

• Probably. The crude normal approximation gives about $0.2588$ for the upper number. – André Nicolas Sep 17 '15 at 18:03

## 1 Answer

It should be a proportion, which it is $phat=0.2103$ plus or minus $0.048$ which gives the interval $[0.1623,0.2585]$ You are close to these values, but don't use percentages.

• Thanks! Any thoughts on the second part? How can connect my answer of the first part to the second part? – Tree Sep 17 '15 at 18:00
• Hmm, I'm not positive on this, but if we are 95% confident that the proportion of members is [0.1623, 0.2585] then perhaps we are 95% confident that the population size is [800,000/0.1623, 800,000/0.2585]. this is because for example our lower bound should be given by 0.1623*(population size)=800,000 , solving for pop. size gives pop. size = 800,000/0.1623. And upper bound same, but pop. size=800,000/0.2585. – MegaboofMD Sep 17 '15 at 18:15
• the upper is bound is smaller than the lower bound? – Tree Sep 17 '15 at 18:30
• those should be reversed. – MegaboofMD Sep 17 '15 at 18:58
• For a FIXED 800,000 club members a larger population proportion (0.2585) would require a smaller total population, and a smaller population proportion (0.1623) would require a larger total population. – MegaboofMD Sep 17 '15 at 19:05