SOLUTION: SUPPOSE THAT A POPULATION PROPORTION IS . 40 AND THAT 80% OF THE TIME YOU DRAW A RANDOM SAMPLE FROM THIS POPULATION , YOU GET A SAMPLE PROPORTION OF 0.35 OR MORE. HOW LARGE A

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Question 1085299: SUPPOSE THAT A POPULATION PROPORTION IS . 40 AND THAT 80% OF THE TIME YOU DRAW A RANDOM SAMPLE FROM THIS POPULATION , YOU GET A SAMPLE PROPORTION OF 0.35 OR MORE. HOW LARGE A SAMPLE WERE YOU TAKING?
Answer by Theo(13342) (Show Source):
You can put this solution on YOUR website!
what i get is this.

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: