Question 1177422
standard error is equal to population standard deviation divided by square root of sample size.


the formula is s = psd / sqrt(ss).


s is the standard error.
psd is population standard deviation.
ss is sample size.


when s = 20 and psd = 500, the formula becomes:


20 = 500 / sqrt(ss)


solve for sqrt(ss) to get:


sqrt(ss) = 500 / 20 = 25


solve for ss to get:


ss = 25^2 = 625


that's the sample size.


the formula for z-score is:


z = (x - m) / s


z is the z-score
x is the raw score
m is the raw mean
s is the standard error.


when (x - m) = 25, and s = 20, the formula becomes:


z = 25 / 20 = 1.25


since the normal distribution curve is symmetric bout the mean, then the confidence interval is z-score of -1.25 to 1.25.


the probability is the area under the normal distribution curve between those 2 z-scores.


that probability is .7887003221.


the probability that the sample mean will be within 25 of the population mean is .7887003221.


this works regardless of the mean.


for example:


assume the mean is 1500.


the z-score formula becomes plus or minus 1.25 = (x - 1500) / 20


solve for the raw score to get:


on the high side, x = 20 * 1.25 + 1500
on the low side, x = 20 * -1.25 + 1500


since 20 * 1.25 is always equal to 25, then you get:


x = 1500 - 25 or x = 1500 + 25.


the sample mean will always be within 25 of the population mean if the z-score is plus or minus 1.25 and the standard error is 20, regardless of what the mean is.


standard error = psd / sqrt(ss)
with a sample size of 625 and a psd of 500, this becomes:
standard error = 500 / sqrt(625) = = 500 / 25 = 20