p = .4
q = 1 - p = .6
s = sqrt(p*q/n) = sqrt(.24/n)
p is the mean proportion which is also called m in the z-score formula
q is 1 minus the mean proportion
s is the standard deviation of the distribution of sample mean proportions
z = (x-m)/s = (.35-.4)/s = -.05/s
z is the z-score
x is the proportion you are testing against the mean proportion.
m is the mean proportion.
s is the standard deviation of the distribution of sample mean proportions.
from z = -.05/s, solve for s to get s = -.05/z
if you get a score of .35 or more 80% of the time, then your alpha has to be .8
that would be 80% of the area under the normal distribution curve is to the right of that score.
this means 20% is to the left of that score.
solve for a z-score that has 20% of the area under the normal distribution curve to the left of that z-score and you get a z-score of -.8416212335
with a z-score of -.8416212335, the formula for s becomes:
s = -.05 / -.8416212335
we know that s is equal to sqrt(.24/n), so we get:
sqrt(.24/n) = (-.05 / -.8416212335)
square both sides of this equation to get:
.24/n = (-.05 / -.8416212335)^2
solve for n to get:
n = .24 / (-.05 / -.8416212335)^2
this results in n = 67.99932487
round this to 68.
your numbers become:
p = .4
q = .6
n = 68
s = sqrt(.24/68) = .0594088526
z-score formula is z = (x-m)/s
this becomes z = (.35 - .4) / .0594088526 which results in:
z = -.8416254112
find the area of the normal distribution curve to the right of this z-score and you get:
alpha = .8000012235 which is slightly above 80%.
visually, it would look like this:
