When Conducting Research into the Efficacy of Drugs to Combat $p = 0.50$

When conducting research into the efficacy of drugs to combat psychological
disorders it is very common to use double blinding as a design strategy. In a double
blind study neither the patient nor the doctor knows whether the patient is receiving
the treatment or the placebo drug. Does double blinding actually work? That is, is
the doctor completely ignorant of the patient's assignment to their treatment group?
a) In a study involving the use of the drugs alprazolam and propranolol, 43 patients
with panic attack disorder and agoraphobia were treated. After completing the
study the physicians involved were asked to guess the treatments. The physicians
correctly guessed the treatments for 30 patients. Determine whether or not the
sample size is large enough to use the standard normal distribution to find the
margin of error. Be sure to show any supporting calculations.
In a different study of double-blinding, the use of lithium as a treatment to counteract
aggressive tendencies in adolescents was under study. In that study the physicians
correctly guessed the treatment in 37 out of 57 patients.
b) Assuming these adolescents and doctors to be representative, estimate the
population proportion of patients for whom the doctor could correctly guess the
treatment.
c) If the physicians were simply guessing about the treatments administered to the patients, $p = 0.50$ would be the proportion of correct guesses. If the physicians were guessing, what would be the standard error of $\hat { p }$ ? d) Based on the given data for the lithium treatment, calculate a $95 \%$ confidence interval for $p$ , the proportion of correct guesses by physicians. e) Suppose the researchers wished to get a more precise estimate of the proportion of correct guesses of treatment when using lithium in this context. What sample size would be needed to obtain a margin of error of $0.05$ with $95 \%$ confidence?