Question

Suppose the manager of a shoe store wants to determine the current percentage of customers who are males. How many customers should the manager survey in order to be 95% confident that the estimated (sample) proportion is within 5 percentage points of the true population proportion of customers who are males?

Z0.10	Z0.05	Z0.04	Z0.025	Z0.01	Z0.05
1.282	1.645	1.751	1.960	2.326	2.576

Answer 1

here margin of error=5%=0.05

with (1-α)*100% confidence margin of error=z(α /2)*SE(p)

with (1-0.05)*100%=95% confidence margin of error=z(0.05 /2)*SE(p)

SE(p)=sqrt(p(1-p)/n)

or, z(0.05 /2)*SE(p)<=0.05

or, 1.96*sqrt(p(1-p)/n)<=0.05

we have to maximize p(1-p) to get the minimum sample size so that margin of error would be less than 5% and p=0.5

or, sqrt(0.5*0.5/n)<=0.05/1.96

or, 0.05/sqrt(n)<=0.0255

or, sqrt(n)>=0.05/0.0255

or, sqrt(n)>=19.61

or, n>=384.55 ( next whole number is 385)

answer is 385