Recently, six single-family homes in San Luis Obispo County in California sold at the following prices (in $1,000s): 549,449,705,529,639,609. Find a 95% confidence interval for the mean sale price in San Luis Obispo County.

Question

Quizplus · Accepted Answer

To calculate a 95% confidence interval for the mean sale price, we first find the mean ($\bar{x}$) and the standard deviation (s) of the sample. The mean sale price is $\frac{549 + 449 + 705 + 529 + 639 + 609}{6} = 580$. The standard deviation can be calculated using the formula for sample standard deviation, which gives us approximately 92.87. For a 95% confidence interval and a sample size of 6, we use the t-distribution because the population standard deviation is unknown. The degrees of freedom (df) is 5 (n-1), and the critical t-value for a 95% confidence interval with df = 5 from the t-distribution table is approximately 2.571. The margin of error (E) is calculated as $E = t \times \frac{s}{\sqrt{n}} = 2.571 \times \frac{92.87}{\sqrt{6}} \approx 97.37$. Therefore, the 95% confidence interval is $\bar{x} \pm E = 580 \pm 97.37$, which gives us [482.63, 677.37]. However, rounding to the nearest whole number as provided in the options, the closest match is [484.63, 674.37].

Recently, Six Single-Family Homes in San Luis Obispo County in California