The formula to find the minimum sample size while testing population proportion is
n = [tex] p (1-p) (\frac{z(\alpha/2)}{ME} ) ^{2} [/tex]
Where p is sample proportion. If there is no information about sample proportion then we take it as 0.5
ME = Margin of error = 0.045
z (α/2) = Critical z score value
This is z score such that P(Z < - z (α/2)) = α/2 and P(Z > z (α/2)) = α/2
α = 1 -confidence level = 1- (96/100) = 0.04
α/2 = 0.04/2 = 0.02
So we have to find z score such that
P(Z < -z (α/2)) = 0.02 and P(Z > z (α/2)) = 0.02
Using z score table to find probability exactly or close to 0.02
Probability value 0.0202 is closest probability value and the corresponding z score is -2.05
It means P(Z < -2.05) =0.02 and P(Z > 2.05) = 0.02
So we have z (α/2) = 2.05
Using all the given values into formula to find sample size
n = 0.5 * (1-0.5) (2.05/0.045)^2
= 0.25 * (45.5556)^2
= 0.25 * 2075.313
n = 518.82
Round the sample size to nearest integer we get
n =519
The minimum sample size is 519
