Sleep apnea is a disorder in which there are pauses in breathing during sleep. People with this condition must wake up frequently to breathe. The article "Postoperative Complications in Patients With Obstructive Sleep Apnea Syndrome Undergoing Hip or Knee Replacement: A Case-Control Study" (R. Gupta, J. Parvizi, et al., Mayo Clinic Proceedings, 2001: \(897-905\) ) reported that in a sample of 427 people 65 and over, 104 of them had sleep apnea.

a. Find a 95\% confidence interval for the proportion of those 65 and over who have sleep apnea.

b. Find a \(99 \%\) confidence interval for the proportion of those 65 and over who have sleep apnea.

c. Find the sample size needed for a \(95 \%\) confidence interval to specify the proportion to within \(\pm 0.03\).

d. Find the sample size needed for a \(99 \%\) confidence interval to specify the proportion to within \(\pm 0.03\).