37. There were 5,317 previously owned homes sold in a western city in the year 2000 . The distribution of the sales prices of these homes was strongly right-skewed, with a mean of \( \$ 206,274 \) and a standard deviation of \( \$ 37,881 \). If all possible simple random samples of size 100 are drawn from this population and the mean is computed for each of these samples, which of the following describes the sampling distribution of the sample mean? (A) Approximately normal with mean \( \$ 206,274 \) and standard deviation \( \$ 3,788 \) (B) Approximately normal with mean \( \$ 206,274 \) and standard deviation \( \$ 37,881 \) (C) Approximately normal with mean \( \$ 206,274 \) and standard deviation \( \$ 520 \) (D) Strongly right-skewed with mean \( \$ 206,274 \) and standard deviation \( \$ 3,788 \) (E) Strongly right-skewed with mean \( \$ 206,274 \) and standard deviation \( \$ 37,881 \)

To determine the sampling distribution of the sample mean, we'll use the **Central Limit Theorem (CLT)**. The CLT states that, for a sufficiently large sample size, the distribution of the sample mean will be approximately normal, regardless of the shape of the population distribution. Given the details: - **Population Mean (\(\mu\))**: \$206,274 - **Population Standard Deviation (\(\sigma\))**: \$37,881 - **Sample Size (\(n\))**: 100 According to the CLT: 1. **Mean of the Sampling Distribution (\(\mu_{\bar{x}}\))**: This is equal to the population mean. \[ \mu_{\bar{x}} = \mu = \$206,274 \] 2. **Standard Deviation of the Sampling Distribution (\(\sigma_{\bar{x}}\))**: This is the population standard deviation divided by the square root of the sample size. \[ \sigma_{\bar{x}} = \frac{\sigma}{\sqrt{n}} = \frac{37,881}{\sqrt{100}} = \frac{37,881}{10} = \$3,788.1 \] Since the sample size is large (\(n = 100\)), the sampling distribution of the sample mean will be approximately **normal**, even though the original distribution is strongly right-skewed. **Answer:** **(A) Approximately normal with mean $206,274 and standard deviation $3,788**