Question 1117965
the elves hits their mark 197 out of 210 times.


that's a success rate of 197/210 = .9380952381.


that's a failure rate of (210 - 197) / 210 = 13/210 = .0619047619.


you can use the binomial distribution formula to solve this, i believe.


in a binomial distribuion:


mean = n * p
standard error of the distribution of sample means = sqrt(n * p * q)


in your problem, this comees out to:


n = 210
p = .9380952381 = success rate
q = 1 - p = .0619047619 = failure rate
m = n * p = 210 * .9380952381 = 197 = mean
s = sqrt(n * p * q) = 3.492168108 = standard error of the distribution of sample means


the critical z-score for a 95% confidence interval is equal to plus or minus 1.959963986.


the formula for z-score is z = (x - m) / s


z is the z-score
x is the raw score
m is the mean
s is the standard deviation / standard error of the distribution of sample means


we know the z-score.
we are looking for the raw score.


on the low side, we get -1.959963986 = (x - 197) / 3.492168108.


solve for x to get x = -1.959963986 * 3.492168108 + 197 = 190.1554763.


on the high side, we get 1.959963986 = (x - 197) / 3.492168108.


solve for x to get x = 1.959963986 * 3.492168108 + 197 = 203.8445237.


what i believe this is saying is that you can expect that the elves would be expected to score between 190 and 204 targets out of 210 for 95% of the battles that they become involved in.


i'm pretty sure this is correct, although i'm not 100% sure this is what your instructor is looking for.


you should verify with other sources, if you can.