Puzzle #222: Statistics Abuse

Is it significant if 52.4% of cases meet a criterion?

Obviously, 524 cases out of 1000 would be 52.4%, but the numbers could be lower. They could be 262 out of 500 or even 131 out of 250. If you allow for rounding, it might be that a far lower number of cases meeting the criterion can be reported as 52.4%.

If you round to the nearest 0.1%, what is the lowest number of cases out of total cases that can be reported as 52.4%?

(52.4% seems so precise and “scientific”, but it is actually an abuse of statistics to report three digits of precision when one has fewer than 1000 cases total. This puzzle will show you how bad it can get; the numbers come from an article I read.)

Submit your answer to Gene Wirchenko <genew@telus.net>. Your answer should be in the form of a proof. That means to show how your answer must be correct. The deadline is Wednesday, September 6, 2017 at noon Pacific Time. I will post the answer shortly after.