p = proportion who like the beer.
q = proportion who don't like the beer = 1 - p.
n = sample size
it's not clear whether you meant 700 people or 500 people.
i'm going to assume you meant 700 people.
if you meant 500 people, you will need to adjust the numbers accordingly.
p = 385/700 = .55
q = 315/700 = .45
n = 700
s = standard error of the proportion = sqrt(p*q/n) = sqrt(.55*.45/700) = .0188034951.
95% confidence interval for the sample percentage would be calculated as follows.
take 100% and subtract 95% from it and then divide it by 2 to get 2.5% on each end of the confidence interval.
this means that 2.5% are outside the confidence interval to the left and 2.5% are outside of the confidence interval to the right.
look up in the z-score table for .025 to the left of the z-score indicated.
you will get a z-score of -1.96.
since the normal distribution tables are symmetric about the mean, this means that your confidence interval is between a z-score of -1.96 to 1.96.
the formula for z-score is z = (x-m)/s
in your problem, x is the raw score of the proportion, m is the mean proportion, and s is the stnadard error of the proportion.
for the low side, you get -1.96 = (x-.55)/.0188034951
solve for x to get x = -1.96*.0188034951+.55 = .5131124043
for the high side, you get 1.96 = (x-.55)/.0188034951
solve for x to get x = 1.96 * .0188034951+.55 = .5868875957.
the 95% confidence interval for the population proportion is .5131124043 to .5868875957.
this should be the solution you are looking for.
this can be shown on the distribution curve as follows.
the first pictures shows that -1.96 to 1.96 z-score give you 95% under the normal distribution curve. this assumes a mean of 0 and a standard deviation of 1.
the second picture shows that the raw scores of -.51... and .58... give you the same 95%. this assumes a mean of .55 and a standard error of .01881....
these pictures show that the translation of z-score to raw score is correct.
here's a reference that discusses some of this.
http://onlinestatbook.com/2/estimation/proportion_ci.html