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 2005. The fear is that house prices will decline and leave homeowners owing more than their homes are worth. PMI Mortgage Insurance estimated that there existed a 50% risk that prices would decline within two years in major metro areas such as San Diego, Boston, Long Island, New York City, Los Angeles, and San Francisco. A survey taken by realtors in the San Francisco area found that 12 out of the 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's proportion is the same as the nationwide proportion of no-money-down loans.
A) 0.0618
B) 0.0124
C) 0.0512
D) 0.0441

Question

Quizplus · Accepted Answer

This probability can be calculated using the binomial distribution formula, where the probability of success (purchasing with no-money-down loans) is 0.43 (43% as given), and the number of trials (first-time buyers sampled) is 20. The question asks for the probability of at least 12 successes (x ≥ 12). This is found by calculating 1 minus the probability of 11 or fewer successes, which involves summing the probabilities of 0 to 11 successes. The exact calculation can be complex and is typically done using statistical software or a binomial probability calculator, which yields the result 0.0618, indicating a relatively low probability of observing at least 12 out of 20 buyers using no-money-down loans if the true proportion is 0.43.

The National Association of Realtors Released a Survey Indicating That