Question

Using R-Studio

We desire to know the probability that a voter supports a controversial rose proposal. From a random sample of 3100, x=1331 voters support the rose proposal. Let p hat be the sample proportion supporting the rose proposal. Answer the following:

a) what is the variance of p hat

b) As a function of p, what is the standard deviation of p hat

c) Calculate p hat

d) Let ptot be the random variable representing the voters in the sample who support the rose proposal. As a function of p, what is the standard deviation of ptot?

e) Calculate a classical 96% confidence interval for p

f) What is the critical value for an approximate classical 94 percent confidence interval for p?

g) What is the length of the 96% confidence interval for p?

h) Assuming the same p hat value, what sample size would have made the 96% confidence interval for p to have a length of .08 or less?

Answer 1

SOlutionA:

Rcode is:

x <- 1331
n <- 3100
phat <- x/n
phat
var_phat <-(phat*(1-phat)/n)
var_phat

Variance=7.903525e-05

b) As a function of p, what is the standard deviation of p hat

sd_phat <-sqrt((phat*(1-phat)/n))
sd_phat

output:

0.008890177

standard deviation= 0.008890177

Solutionc:

c) Calculate p hat

x <- 1331
n <- 3100
phat <- x/n

phat=0.4293548

d) Let ptot be the random variable representing the voters in the sample who support the rose proposal. As a function of p, what is the standard deviation of ptot?

ptot=sqrt(p*(1-p)/n)

=sqrt(0.4293548*(1-0.4293548)/3100)

=0.0089

e) Calculate a classical 96% confidence interval for p

install library PropCIs in R

Rcode:

library(PropCIs)
require(PropCIs)

exactci(1331, 3100, conf.level=0.96)

Output:

Data:  

96 percent confidence interval:
0.4110156 and 0.4478390