Sleep researchers theorize that 25 of the general population

Sleep researchers theorize that 25% of the general population suffers from obstructive sleep apnea. Researchers found that 124 of 159 emergency room nurses suffered from obstructive sleep apnea. Calculate an interval to test the hypotheses at =.001.

Solution

First, we get the point estimate of the proportion, p^,              
              
p^ = x / n =    0.779874214          
              
Also, we get the standard error of p, sp:              
              
sp = sqrt[p(1 - p) / n] =    0.034340141          
              
Now, for the critical z,              
alpha/2 =   0.0005          
Thus, z(alpha/2) =    3.290526731          
Thus,              
              
lower bound = p^ - z(alpha/2) * sp   0.666877062          
upper bound = p^ + z(alpha/2) * sp =    0.892871366          
              
Thus, the confidence interval is              
              
(   0.666877062   ,   0.892871366   )

As 0.25 is not part of this interval, then we conclude that at 0.001 significance, there is no significant evidence to say that 25% of the general population suffers from obstructive sleep apnea.