A market research firm conducts telephone surveys with a 50% historic response rate. suppose a sample of 400 is taken. what is the probability that the sample proportion will be at least 0.475?

N=400

sample proportion > =.475 = > n>=.475N=190

mean response rate of population, p=0.5

Problem satisfies criteria for binomial distribution, where

P (x>=190) = sum P (x=i), i=190 to 400

This can be obtained from binomial distribution tables, calculators, or software as

P (x>=190) = 1-P (x<190) = 1-0.14685=0.85315

Alternatively, normal approximation can be used.

mean=Np=400*.5=200

variance=Npq=400 (.5) (1-.5) = 100

standard deviation=sqrt (variance) = sqrt (100) = 10

Calculate P (X>=190) = 1-P (X<190)

X with continuity correction = 189.5

Z = (X-mean) / standard deviation = (189.5-200) / 10=-1.05

P (z
hence

P (X>=190) = 1-P (X<190) = 1-P (z<-1.05) = 1-0.14686=0.85314 (approximately)

almost same as the exact binomial solution.