Question

In an Introduction to Business Management class If the true (population) proportion of students in college residential apartments p was found to be 0.65, what is the probability that a sample of size n = 200 will have a sample proportion less than 0.55?

Answer 1

Sampling distribution of p̂ is approximately normal if np >=10 and n (1-p) >= 10
n * p = 200 * 0.65 = 130
n * (1 - p ) = 200 * (1 - 0.65) = 70

Mean = = p = 0.65
Standard deviation = = 0.033727

X ~ N ( µ = 0.65 , σ = 0.033727 )
P ( X < 0.55 )
Standardizing the value
Z = ( X - µ ) / σ
Z = ( 0.55 - 0.65 ) / 0.033727
Z = -2.965
P ( ( X - µ ) / σ ) < ( 0.55 - 0.65 ) / 0.033727 )
P ( X < 0.55 ) = P ( Z < -2.965 )
P ( X < 0.55 ) = 0.0015