The other day I heard the business reporter on the radio say "Advancing stocks led declining stocks by 7 to 5." That is, 7/12 of all stocks went up for the day, and 5/12 went down.
Now, there are thousands of stocks in the market, so 7/12 is most likely just an approximation to the actual number. My question is, how good an approximation is it? You have enough information to answer the question if you make one reasonable assumption.
up, and estimate proportion P of the pouulation that went up by p=n/N.
Then if N is large enough n is a random variable with a binomial distribution
mu = NP
and standard deviation:
sigma = sqrt(NP(1-P))
so p is a RV with SD sigma/N.
If instead we look at the entire population of size N, and find n went up then
there is no error in putting p=n/N as the proportion that went up.
Hmm, not going in the direction I intended so let me clarify a little.
Given a real number (not necessarily rational) between 0 and 1, you want to find the "best" rational approximation to it. You might say for example "half of the voters in the US are Democrats." Now the actual proportion is unlikely to be exactly .50000..., but maybe it's really .49653..., so most people would say 1/2 is the best rational approximation you're going to come up with.
So my original question can be restated, If 7/12 is the best rational approximation to a particular real number, what can you tell me about that number? How good an approximation is 7/12? There's already enough information to solve the problem if you make one reasonable assumption.
*)We define it the most accurate** rational with a given smallest denominator
**)We define it that |x-p/q|<e where e is the error.
There is no way to get an irrational number from the problems you presented.
Let's say that 33 out of 100 random people were guys. That would mean that the percentage of guys is 33.33333333333333333333333333333333333333333333333 333333333333333333333333333333333
or you could just say that it's 2/3 guys.
In this case 2/3 is the exact number, whereas the decimal is not.
A fraction is either just as good or better than a decimal.
the best rational approximation to an irrational number, but here we are
talking about being given a rational approximation to a (not-necessarily
irrational number - in fact it has to be rational as there are a finite number of
stocks) what range of proportions are consistent with that rational approximation.
In these terms we see that there is in fact no answer because we don't know
what psychological factors determined how the number quoted was derived,
i.e. what the rounder/approximater considered an socially acceptable
In everyday life, if somebody uses 7/12 as an approximation, it must be because there is no fraction with a lower denominator that gives a better approximation.
That means the number is closer to 7/12 (=.58333...) than it is to 4/7 (=.57142...) or 3/5 (=.6). So the number must be between .57738... and .59166..., or an error of less than 1% in either direction.