Question

According to a recent report, it was found that 50.3% of residents in Cuyahoga County, Ohio are registered to vote. Which of the following is more likely?

Select one:

A) We take a random sample of 100 people from this county and find that the proportion is below 45%

B) We take a random sample of 400 people from this county and find that the proportion is below 45%

C) We take a random sample of 1000 people from this county and find that the proportion is below 45%

D) We take a random sample of 10000 people from this county and find that the proportion is below 45%

E) We have no basis for predicting which is more likely to have a proportion below 45%.

Answer 1

a)
E(Phat) = p =    50.30%  
V(Phat) = pq/n = 0.503*(1-0.503)/100 = 0.00249991
sd= sqrt(var)   0.0500

P( X<0.45) = ?
I know that, z = (X-mean)/(sd)  
z1 = (0.45-0.503)/0.049999) = -1.0600
hence,  
P( X<0.45) = P(Z<-1.06)
NORMSDIST(-1.06) = 0.1446

b)
E(Phat) = p = 50.30%  
V(Phat) = pq/n = 0.503*(1-0.503)/400 = 0.000624978
sd= sqrt(var)   0.0250
P( X<0.45) = ?
I know that, z = (X-mean)/(sd)  
z1 = (0.45-0.503)/0.024999) = -2.1200
hence,  
P( X<0.45) = P(Z<-2.12)
NORMSDIST(-2.12) = 0.0170

c)
E(Phat) = p = 50.30%  
V(Phat) = pq/n = 0.503*(1-0.503)/1000 = 0.000249991
sd= sqrt(var) = 0.0158
P( X<0.45) = ?
I know that, z = (X-mean)/(sd)  
z1 = (0.45-0.503)/0.015811103693291)   -3.3521
hence,  
P( X<0.45)   =P(Z<-3.3521)
NORMSDIST(-3.3521) =   0.0004

d)
E(Phat) = p = 50.30%  
V(Phat) = pq/n = 0.503*(1-0.503)/10000 = 0.000025
sd= sqrt(var) = 0.0050
P( X<0.45) = ?
I know that, z = (X-mean)/(sd)  
z1 = (0.45-0.503)/0.005) = -10.6000
hence,  
P( X<0.45) = P(Z<-10.6)
NORMSDIST(-10.6) = 0.0000

HENCE OPTION A IS CORRECT