Probability of a Proportion

Question

Suppose that 78% of all dialysis patients will survive for at least 5 years.

In a simple random sample of 100 new dialysis patients, what is the probability that the proportion surviving for at least five years will exceed 80%, rounded to 5 decimal places?

Answer 1

want to find P(phat > 0.8) when p = 0.78, n = 100

z = ( (phat - p)/sqrt(p*(1-p)/n) is distributed Normal with mean 0 and standard deviation 1

z = ((0.8 - 0.78)/sqrt(0.78*0.22/100) = 0.48

P(phat > 0.8) = P(z > 0.48) = 1 - P(z < 0.48) =

(from standard normal probabilities table) 1 - 0.6844 = 0.3156