Question

Based on past experience, a bank believes that 8 % of the people who receive loans will not make payments on time. The bank has recently approved 200 loans.

A. What must be true to be able to approximate the sampling distribution with a normal model? (Hint: think Central Limit Theorem) Assumptions:  

B. What are the mean and standard deviation of this model?
mean =
standard deviation (accurate to 3 decimal places) =

C. What is the probability that over 10% of these clients will not make timely payments?

Answer 1

a)

Assumptions:probability of one client who not make payments on time is independent of other client

np=200*8%=16≥10

nq=200*92%=184≥10

so, according to central limit theorem , sampling distribution can be approximated with a normal model

b)

population proportion ,p=   0.08
n=   200

mean=np=16
  
std dev= , SE = √( p(1-p)/n ) =    0.019

c)

p̂ =0.10

Z=( p̂ - p )/SE=   1.043      
P ( p̂ >    0.10 ) =P(Z > ( p̂ - p )/SE) =  
          
=P(Z >   1.043   ) =    0.1486

so, 14.86% probability that over 10% of these clients will not make timely payments