Final answer:
The sampling distribution for the proportion of homeowners with ARMs in danger of defaulting is normally distributed with a mean of 0.0183 and a standard deviation calculated by the formula σς = √ [0.0183(1 - 0.0183) / 214], based on a sample of 214 homeowners.
Step-by-step explanation:
The student asked about the sampling distribution of the sample proportion of homeowners with adjustable-rate mortgages in danger of defaulting based on a sample of 214 homeowners. The Mortgage Lenders Association reported an early foreclosure rate of 1.83%. To describe the sampling distribution, we assume the population proportion (π) to be 0.0183, and since the sample size (n) is 214, which is large enough, we can approximate the sampling distribution of the sample proportion (ς) by a normal distribution due to the Central Limit Theorem. The mean of the sampling distribution will be equal to the population proportion (π), which is 0.0183, and the standard deviation (σς) is determined by the formula σς = √ [π(1 - π) / n]. Therefore, the standard deviation of the sampling distribution will be σς = √ [0.0183 × (1 - 0.0183) / 214].