Question 1202132
population mean is 1500.
population standard deviation is 289.
margin of error is less than or equal to plus or minus 50.



z-score formula is z = (x-m) / s
z is the z is the critical z-score.
(x-m) is the margin of error.
s is the standard error.
standard error = standard deviation / sqrt(n)
sd is the standard deviation.
n is the sample size.
z-score formula becomes z = (x-m) / (sd / sqrt(n))
simplify to get z = (x-m) / sd * sqrt(n).


solve for sqrt(n) to get sqrt(n) = z / (x-m) * sd *****


once you have sqrt(n), confirm by solving for (x-m) to get (x-m) = z * sd / sqrt(n) *****


critical z-score at 85% confidence interval is plus or minus z = 1.439531471.
the .15 alpha is divided by 2 to get .075 alpha on each side of the confidence interval.


when z = that and (x-m) = 50 and sd = 289, solve for sqrt(n) to get:
sqrt(n) = 8.320491902.


solve for (x-m) to get (x-m) = 1.439531471 * 289 / 8.320491902. = 50.
this confirms the margin of error is what you want it to be.


n = sqrt(n)^2 = 8.320491902^2 = 69.23058549.
n has to be an integer to set it to the next higher integer = 70.
sqt(n) is now equal to sqrt(70) = 8.366600265.


solve for (x-m) to get (x-m) = 1.439531471 * 289 / 8.366600265 = 49.72444983.
the mrgin of error is less than 50 as desired.


since s = sd / sqrt(n), you get s = 289 / 8.366600265 = 34.54210681.


when m = 1500 and s = 34.54210681 and z = 1.439531471, z-score formula becomes:
1.439531471 = (x - 1500) / 34.54210681.
solve for x to get:
x = 1.439531471 * 34.54210681 + 1500 = 1549.72445.
that's on the high side of the 85% confidence interval.
on the low side, you get -1.439531471 * 34.54210681 + 1500 = 1450.27555.


your margin of error is less than 50, as it should be.


once you found sample size to the  next higher integer, you solve for standard error to get s = 289 / sqrt(sample size).  
you then use that value of standard error in the z-score formula to find x.


here's what it looks like using the z-score calculator at <a href = "https://davidmlane.com/hyperstat/z_table.html" target = "__blank">https://davidmlane.com/hyperstat/z_table.html</a>


<img src = "http://theo.x10hosting.com/2023/050701.jpg">


let md know if you have any questions.
theo