According to the local real estate board, the average number of days that homes stay on the market before selling is 78.4, with a standard deviation equal to 11 days. A prospective seller selected a random sample of 36 homes from the multiple listing service. Above what value for the sample mean should 95 percent of all possible sample means fall?
A) about 75.4 days
B) about 81.4 days
C) about 96.5 days
D) about 60.3 days
E) about 79.3 days

Question

Quizplus · Accepted Answer

To find the value above which 95% of all possible sample means fall, we use the formula for the lower bound of a confidence interval for the mean, which is $\bar{x} - Z\frac{\sigma}{\sqrt{n}}$, where $\bar{x}$ is the sample mean, $Z$ is the Z-score corresponding to the confidence level, $\sigma$ is the population standard deviation, and $n$ is the sample size. For a 95% confidence level, the Z-score is approximately 1.96. Given $\sigma = 11$ days and $n = 36$, the standard error of the mean ($\frac{\sigma}{\sqrt{n}}$) is $11/\sqrt{36} = 11/6 = 1.833$. The margin of error is $1.96 \times 1.833 \approx 3.6$. Since we're looking for the value above which 95% of sample means fall, we subtract this margin of error from the population mean: $78.4 - 3.6 = 74.8$, which is approximately 75.4 days. This calculation assumes a normal distribution of sample means according to the Central Limit Theorem.

According to the Local Real Estate Board, the Average Number