The national association of realtors released a survey indic

The national association of realtors released a survey indicating that a surprising 43% of first time home buyers purchased their homes with no money down loans during the time before the housing bubble collapsed. Many house prices declined, leaving, homeowners owing more than their homes are worth. Even today some forecasters estimate their exists a 50% risk that prices will decline within two years in major metro areas such as San Diego, Boston, Long Island, New York City, Los Angeles, and San Francisco.

A. A survey taken by realtors in the San Francisco area found that 12 out of 20 first time home buyers sampled purchased their home with no money down loans. Calculate the probability that at least 12 in a sample of 20 first time buyers would take out no money down loans if San Francisco proportion is the same as the nationwide proportion of no money down loans.

B. Determine the probability requested in part a if the nationwide proportion is 0.53.

C. Determine the probability that between 8 and 12 of a sample of 20 first time home buyers would take out no money down loans if the 43% value applies?

Solution

Normal Distribution
Mean ( np ) =8.6
Standard Deviation ( npq )= 20*0.43*0.57 = 2.214
Normal Distribution = Z= X- u / sd                   
a)
P(X < 12) = (12-8.6)/2.214
= 3.4/2.214= 1.5357
= P ( Z <1.5357) From Standard NOrmal Table
= 0.9377                  
P(X>=12) = 1 - P(X < 12) = 1 - 0.9377 = 0.0623
b)
P(X < 12) = (12-10.6)/2.232
= 1.4/2.232= 0.6272
= P ( Z <0.6272) From Standard NOrmal Table
= 0.7347                  
P(X>=12) = 1 - P(X < 12) = 1 - 0.7347 = 0.2653
c)
To find P(a < = Z < = b) = F(b) - F(a)
P(X < 8) = (8-8.6)/2.214
= -0.6/2.214 = -0.271
= P ( Z <-0.271) From Standard Normal Table
= 0.39319
P(X < 12) = (12-8.6)/2.214
= 3.4/2.214 = 1.5357
= P ( Z <1.5357) From Standard Normal Table
= 0.93769
P(8 < X < 12) = 0.93769-0.39319 = 0.5445