Question 1162115
population mean = m = 100
population standard deviation = sd = 17
sample size = n
standard error = s = sd / sqrt(n)
z-score formula is z = (x - m) / s
at 95% confidence limit, critical z-score is plus or minus 1.959963986.
you want (x - m) to be equal to plus or minus 6.
work with the upper limit first.
(x - m) = (106 - 100) = 6
z-score formula becomes z = 6 / s
solve for s to get s = 6 / z
since s = sd / sqrt(n), you get sd / sqrt(n) = 6 / z
solve for sqrt(n) to get sqrt(n) = z * sd / 6
since z = 1.959963986 and sd = 17, you get sqrt(n) = 1.959963986 * 17 / 6 = 5.553231204
solve for n to get n = sqrt(n)^2 = 30.8383778
that's your sample size required to get (x-m) equal to plus or minus 6 at 95% confidence limit.
to confirm, use s = 17 / sqrt(n) = 3.061280739 in your calculations.
at 95% confidence limits, your lower and upper z-score formulas become:
-1.959963986 = (x - 100) / 3.061280739 for lower z-score.
1.959963986 = (x - 100) / 3.061280739 for upper z-score.
solve for lower x and upper x to get:
lower x = -1.959963986 * 3.061280739 + 100 = 94
upper x = 1.959963986 * 3.061280739 + 100 = 106
your sample size needs to be equal to 30.8383778 is your answer.
since sample size needs to be an integer, then use sample size = 31.
that will ensure that (x - m) will be less than or equal to 6.
visually, this will look like this.


<img src = "http://theo.x10hosting.com/2020/071001.jpg" >