A cancer with poor prognosis, a three-year mortality of $85\%$, is studied. A new mode of chemotherapy is to be evaluated. Suppose that when testing at the 0.05 significance level, one wishes to be $95\%$ certain of detecting a difference if survival has been increased to $50\%$ or more. The randomized clinical trial will have equal numbers of people in each group. How many patients should be randomized?
My attempt:
p = 0.85 Z = 1.96 for 0.05 significance level As it says $95\%$ sure, this implies the error cannot be more than 5% , thus Margin of error (E ) = 0.05 Thus sample size = $p * (1-p) * (Z / E)^2$ On plugging the values we get: Sample size = $0.85*0.15*(1.96/0.05)^2$ Sample size = 195.92 = 196
Am I missing something?