91Ó°ÊÓ

When an economist wants to estimate a population mean, she must determine the appropriate sample size. This is crucial as it impacts both accuracy and confidence in her results. The formula used is\[n = \left(\frac{Z(\alpha/2) \times \sigma}{E}\right)^2\]where:

• \(n\) is the sample size,
• \(Z(\alpha/2)\) is the Z-score associated with the confidence level,
• \(\sigma\) is the standard deviation of the population,
• \(E\) is the margin of error.

To begin, choose a desired confidence level, like 90%. Calculate the sample size with the non-rounded result and remember to round up. This ensures she's not underestimating the sample size, providing more accurate and reliable estimates of the population mean.

Standard deviation (\(\sigma\)) is a measure that indicates the amount of variation or dispersion in a set of values. In this exercise, it's given as $31,500 for the sale prices of houses. This denotes how much individual house prices typically deviate from the average sale price. A larger standard deviation means more spread out data, while a smaller one signifies values that are closer to the mean.

Understanding this helps in interpreting the reliability of the data. In sampling, a higher standard deviation would generally require a larger sample size for precise estimates, to ensure the sample sufficiently represents the diverse population.
Standard deviation also plays a crucial role in calculating the confidence interval, affecting both the width of the interval and the necessary sample size.

The margin of error (\(E\)) is the range within which the true population parameter is expected to lie. This exercise targets a margin of error of \(\$3500\). The smaller the margin of error, the more precise the estimate—but achieving this often requires a larger sample size.

Selecting an appropriate margin of error involves a trade-off between precision and practicality. A smaller margin means increased data collection and analysis efforts. It is also determined by how critical precision is for the decision or policy formulation based on the data.
In essence, the margin of error helps in defining the "room for error" in your estimates, giving you a buffer around the population mean that accounts for natural sample variability.