An automobile manufacturer would like to know what proportio

An automobile manufacturer would like to know what proportion of its customers are not satisfied with the service provided by the local dealer. The customer relations department will survey a random sample of customers and compute a 90% confidence interval for the proportion who are not satisfied.

(a) Past studies suggest that this proportion will be about 0.27. Find the sample size needed if the margin of the error of the confidence interval is to be about 0.02.
(You will need a critical value accurate to at least 4 decimal places.)
Sample size:

(b) Using the sample size above, when the sample is actually contacted, 30% of the sample say they are not satisfied. What is the margin of the error of the confidence interval?
MoE:  

Solution

An automobile manufacturer would like to know what proportion of its customers are not satisfied with the service provided by the local dealer. The customer relations department will survey a random sample of customers and compute a 90% confidence interval for the proportion who are not satisfied.

(a) Past studies suggest that this proportion will be about 0.27. Find the sample size needed if the margin of the error of the confidence interval is to be about 0.02.
(You will need a critical value accurate to at least 4 decimal places.)
Sample size:

Sample size

P=0.27

For 90%, z=1.6449

d=0.02

Sample size = (z²*p*(1-p))/d²

= (1.6449²*0.27*0.73)/0.02²

=1333.15

The sample size required= 1334

(b) Using the sample size above, when the sample is actually contacted, 30% of the sample say they are not satisfied. What is the margin of the error of the confidence interval?
MoE:  

For 90%, z=1.6449

MOE= z*sqrt(p*(1-p)/n))

=1.6449*sqrt(0.3*0.7/1334)

=0.0206