Question 1197734
population standard deviation is 92.
you want to get a margin of error of 13.
standard error = standard deviation / square root of sample size.
s = 92 / sqrt(n)
n is the sample size
you want margin of error to be equal to 13.
z-score formula is z = (x - m) / s
z is the z-score
x is the raw score
m is the mean
s is the standard error.
you want (x - m) to be equal to plus or minus 13.
because the normal distribution is symmetric about the mean, you only need to find one side and will automatically get the other side with a change of sign.
at 95% confidence interval, the critical z-score on the high side of the confidence interval is equal to z = 1.96.
z-score formula becomes:
1.96 = 13 / (92 / sqrt(n))
multiply both sides of this equation by (92 / sqrt(n)) to get:
1.96 * 92 / sqrt(n) = 13
multiply both sides by sqrt(n) and divide both sides by 13 to get:
1.96 * 92 / 13 = sqrt(n)
solve for sqrt(n) to get:
sqrt(n) = 13.87076923.
when sqrt(n) = that, s (standard error) = 92 / 13.87076923 = 6.632653061.
critical z-score formula becomes:
z = 13 / 6.632653061 = 1.96, confirming you will get the 95% confidence interval of between z = -1.96 and 1.96 (you only derived the high side z-score; the low side z-score is the same with a change of sign)
you can use the z-score formula to see if what you got is correct.
i used an online z-score calculator at <a href = "https://davidmlane.com/hyperstat/z_table.html" target = "_blank">https://davidmlane.com/hyperstat/z_table.html</a>
the mean can be anything and the standard deviation is 92 and the standard error is 6.632653061 and the result will be a margin of error of 13.
here are some examples.
the first is using the z-score with a mean of 0 and a standard deviation of 1.
the rest are using a mean of 100, 200, 5000, with a standard deviation of 6.632653061.
note that the calculator says standard deviation, but when you are using the calculator with the mean of a sample, standard deviation is really standard error.
in all of the examples except the first, the standard error was 6.632653061.
this was truncated by the calculator to 6 decimal digits.
i had no control over that.


<img src = "http://theo.x10hosting.com/2022/110601.jpg">


<img src = "http://theo.x10hosting.com/2022/110602.jpg">


<img src = "http://theo.x10hosting.com/2022/110603.jpg">


<img src = "http://theo.x10hosting.com/2022/110604.jpg">


regardless of the mean, the margin of error will be 13 as long as the standard error is equal to 6.632653061.


since sqrt(n) = 13.87076923, then n = that squared = 192.3982391.
the margin of error will be less than or equal to 13 when the sample size is greater than or equal to 192.3982391.
since sample size needs to be an integer, then minimum sample size will be 193.
this will result in a margin of error slightly less than 13.


let me know if you have any questions.
theo