Some real estate specialists estimate that the length of time people live in a house has a mean of 10 years and a standard deviation of 3 years.A random sample of 200 families was chosen and surveyed.Let $\overline{y}$ represent the mean number of years that those families had lived in their house.Describe the sampling distribution model of this mean.

A) Binom(10,3)
B) N(10,1.5)
C) N(10,3)
D) N(10,0.12)
E) There is not enough information to describe the distribution.

The sampling distribution of the mean for a large sample size (n=200) can be approximated by a normal distribution due to the Central Limit Theorem. The mean of the sampling distribution ( ${\mu }_{\overline{y}}$ ) is equal to the population mean ( $\mu$ ), which is 10 years. The standard deviation of the sampling distribution ( ${\sigma }_{\overline{y}}$ ) is equal to the population standard deviation ( $\sigma$ ) divided by the square root of the sample size ( $n$ ), which is $\frac{3}{\sqrt{200}}$ , approximately equal to 0.21, not 0.12 as stated. However, given the options, D is the closest to the correct calculation, acknowledging a mistake in the calculation presented in the answer choices.