Question 1182680
critical z-score for 95% confidence level is plus or minus 1.96.
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.
when m is 100, formula becomes:
plus or minus 1.96 = (x - 100) / s
formula for standard error is:
s = standard deviation divided by square root of sample size.
with a standard deviation of 12, this becomes:
s = 12 / sqrt(sample size)
when x = 102 and s = 12 / sqrt(sample size), the formula becomes:
1.96 = (102 - 100) / (12 / sqrt(sample size).
this is equivalent to:
1.96 = (102 - 100) * sqrt(sample size) / 12
simplify to get:
1.96 = 2 * sqrt(sample size) / 12
multiply both sides of this equation by 12 and divide both sides of this equation by 2 to get to get:
1.96 * 12 / 2 = sqrt(sample size)
solve for sqrt(sample size) to get:
sqrt(sample size) = 1.96 * 12 / 2 = 11.76
solve for s (standard error) to get:
s = standard deviation / sqrt(sample size) = 12 / 11.76 = 1.020408163
z=score formula becomees 1.96 = (x - 100) / 1.020408163
solve for x to get:
x = 1.020408163 * 1.96 + 100 = 102.
this is what you wanted since 102 - 100 = 2.
when z = -1.96, the formula becomes:
-1.96 = (x - 100) / 1.020408163
solve for x to get:
x = -1.96 * 1.020408163 + 100 = 98.
this is what you wanted since 98 - 100 = -2.
your 95% interval is between 98 and 102.
the sample size required for this is 11.76 squared = 138.2976.
if you need an integer answer, then you would round up to 139.
here's what it looks like on a graph.


<img src = "http://theo.x10hosting.com/2021/071201.jpg" >


SD on the graph standard for standard deviation which would be the standad deviation of the population or the standard error of the sample of a given size.
in this case, we are dealing with the standard error of a sample of size of 138.2976 which is equal to the standard deviation of the population divided by the square root of 138.2976 which is equal to 1.020408163.
there i some rounding with this calculator for input and outputs.   the results are accurate enough for normal problem requirements.