The You Want It Motor Company has found a problem with the v

The You Want It Motor Company has found a problem with the vanity mirrors in their Lukatmia sedans. They checked 854 cars and found the problem in 259 of them. What is the 95% confidence interval for the proportion of cars that have the problem?      

A             0.0902 to 0.1323

B             0.2628 to 0.3438

C             0.2724 to 0.3341

D             0.2774 to 0.3292

E              0.3081 to 0.4255

Solution

Note that              
              
p^ = point estimate of the population proportion = x / n =    0.303278689          
              
Also, we get the standard error of p, sp:              
              
sp = sqrt[p^ (1 - p^) / n] =    0.015729741          
              
Now, for the critical z,              
alpha/2 =   0.025          
Thus, z(alpha/2) =    1.959963985          
Thus,              
              
lower bound = p^ - z(alpha/2) * sp =   0.272448964          
upper bound = p^ + z(alpha/2) * sp =    0.334108413          
              
Thus, the confidence interval is              
              
(   0.272448964   ,   0.334108413   ) [ANSWER, C]