Suppose 35% of homes in a Miami, Florida, neighborhood are under water (in other words, the amount due on the mortgage is larger than the value of the home). There are 160 homes in the neighborhood and 30 of those homes are owned by your friends. What is the probability that fewer than 30% of your friends' homes are under water?

Question

Quizplus · Accepted Answer

This is a binomial distribution problem. Let X be the number of your friends' homes that are under water, then X follows a binomial distribution with parameters n = 30 and p = 0.35 (since 35% of homes in the neighborhood are under water). We want to find P(X < 0.3 * 30) = P(X < 9). Using the binomial distribution formula, we have:

P(X < 9) = Σ_{k=0}^8 (30 choose k) * 0.35^k * (1-0.35)^(30-k)

This is a cumbersome calculation, but can be done using a calculator or software. Alternatively, we can use a normal approximation to the binomial distribution, which gives a good approximation for large n and not-too-extreme values of p. In this case, the mean of X is 30 * 0.35 = 10.5 and the standard deviation is sqrt(30*0.35*(1-0.35)) = 2.8. Then, we can standardize X and use a normal distribution table or calculator to find:

P(X < 9) = P(Z < (8.5-10.5)/2.8)

where Z is a standard normal variable with mean 0 and standard deviation 1. Evaluating this probability gives approximately 0.2627. Therefore, the best choice is A.

Suppose 35% of Homes in a Miami, Florida, Neighborhood Are