A random sample of size n 130 yielded p hat 8 Is the sample

A random sample of size n = 130 yielded p hat .8

Is the sample large enough to construct a valid confidence interval for p? Explain.

Construct a 95% confidence interval for p.

Solution

As

np = 130*0.8 = 104 > 10
nq = 130*0.2 = 26 > 10

then it is large enough.

Note that              
              
p^ = point estimate of the population proportion = x / n =    0.8          
              
Also, we get the standard error of p, sp:              
              
sp = sqrt[p^ (1 - p^) / n] =    0.035082321          
              
Now, for the critical z,              
alpha/2 =   0.025          
Thus, z(alpha/2) =    1.959963985          
Thus,              
Margin of error = z(alpha/2)*sp =    0.068760085          
lower bound = p^ - z(alpha/2) * sp =   0.731239915          
upper bound = p^ + z(alpha/2) * sp =    0.868760085          
              
Thus, the confidence interval is              
              
(   0.731239915   ,   0.868760085   ) [ANSWER]