SOLUTION: A survey reports it results by starting that standard error of the mean to be is 20. The population standard deviation is 500. 1:How large is the sample used in this surveying?

Question 1177422: A survey reports it results by starting that standard error of the mean to be is 20.
The population standard deviation is 500.
1:How large is the sample used in this surveying?
2: What is the probability that the sample mean will be within 25 of the population mean?

Answer by Theo(13342) (Show Source): You can put this solution on YOUR website!
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
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