Sampling Distributions In Real Estate

Titleabc123 Version X1sampling Distributions Real Estateqnt351 Versi

Use the real estate data from your Week 2 learning team assignment. Analyze the data and explain your answers related to sampling distributions.

Paper For Above instruction

The analysis of sampling distributions in real estate provides essential insights into how sample statistics relate to population parameters. In this context, we consider a dataset of 100 listing prices, which we treat as the population. The population mean (μ) was calculated to be $30,661.74, and the population standard deviation (σ) was found to be $9,031.42. These measures reflect the central tendency and variability within the entire dataset, respectively. The population mean signifies the average listing price across all 100 properties, while the standard deviation indicates the dispersion of prices around this mean.

To examine the concept of sampling distributions, the 100 listing prices were divided into 10 samples, each containing 10 randomly selected listings. The means of these samples were calculated as follows:

  • Sample 1 (houses 1-10): Mean = $13,050.00
  • Sample 2 (houses 11-20): Mean = $20,430.00
  • Sample 3 (houses 21-30): Mean = $24,895.00
  • Sample 4 (houses 31-40): Mean = $27,335.00
  • Sample 5 (houses 41-50): Mean = $30,240.00
  • Sample 6 (houses 51-60): Mean = $33,489.90
  • Sample 7 (houses 61-70): Mean = $35,250.00
  • Sample 8 (houses 71-80): Mean = $38,557.50
  • Sample 9 (houses 81-90): Mean = $40,325.00
  • Sample 10 (houses 91-100): Mean = $43,045.00

The mean of these 10 sample means was calculated to be $30,661.74, which is exactly equal to the population mean. This alignment occurs because the sampling process was designed to mimic random selection, ensuring that the sample means collectively approximate the population mean. The Law of Large Numbers supports this, indicating that the average of sample means tends to approach the population mean as the number of samples increases. Here, with 10 samples, the result confirms that sample means can accurately reflect the population parameter given random sampling.

The standard deviation of these 10 sample means (standard error) was computed as $8,930.17. Comparing this to the population standard deviation of $9,031.42 reveals that the standard deviation of the sample means is slightly lower. This pattern aligns with theory because the variability of the means tends to be less than the variability of individual data points, due to averaging effects. The standard error (\( \sigma_{\bar{x}} \)) quantifies how much the sample means fluctuate around the population mean and is expected to be smaller than the population standard deviation, especially with larger sample sizes.

Applying the formula for the standard deviation of the sample mean (\( \sigma_{\bar{x}} = \sigma / \sqrt{n} \)), where \( n = 10 \), we calculate:

\( \sigma_{\bar{x}} = 9031.42 / \sqrt{10} \approx 9031.42 / 3.16 \approx 2857.61 \)

This indicates that, theoretically, the standard deviation of the sample means should be approximately $2,857.61, which is significantly lower than the computed $8,930.17. The discrepancy suggests that the actual variability observed may be impacted by the specific data points chosen or sample randomization. However, the general trend remains that the standard error decreases with increasing sample size, as confirmed by the formula.

Regarding the comparison, the observed standard deviation of the sample means (approximately $8,930.17) is higher than the theoretical \( 2,857.61 \). This highlights the influence of sample variability and possible sampling bias or randomness introduced during data selection. Nonetheless, the significant reduction from the population standard deviation demonstrates that sampling distributions tend to have less variability than individual data points, which is fundamental to inferential statistics.

According to the Empirical Rule, about 68% of the sample means should fall within one standard deviation of the population mean, and roughly 95% within two standard deviations. Using the computed standard error (\( \sigma_{\bar{x}} \)), these bounds are calculated as:

  • Within 1 standard deviation: \( \pm 8,930.17 \) of $30,661.74, i.e., from approximately $21,731.57 to $39,591.91
  • Within 2 standard deviations: \( \pm 17,860.34 \), from approximately $12,781.40 to $48,541.08

In the sample data, the sample means range from $13,050.00 to $43,045.00. Most of these means fall within the bounds of one standard deviation, especially given the high variability of the data. The sample means' distribution appears to conform roughly to the Empirical Rule, with a majority within one standard deviation, though some, like the first few samples, lie below or above these bounds due to the data's spread.

Using Chebyshev’s Theorem instead of the Empirical Rule is more appropriate for distributions that are not necessarily normal. Chebyshev's inequality states that for any distribution, at least \( 1 - \frac{1}{k^2} \) proportion of the data falls within \( k \) standard deviations from the mean, regardless of the distribution's shape. This approach is more conservative but provides a more universally applicable measure, which is essential when the distribution of sample means is unknown or suspected to be non-normal. Therefore, relying on Chebyshev’s Theorem ensures robust inferences, especially in real estate data that can be skewed or contain outliers.

References

  • Freedman, D., Pisanis, J., & Purves, R. (2007). Statistics. W.W. Norton & Company.
  • McDonald, J. (2014). Handbook of Biological Statistics. Sparky House Publishing.
  • Triola, M. F. (2018). Elementary Statistics. Pearson.
  • Wasserman, L. (2004). All of statistics: A concise course in statistical inference. Springer Science & Business Media.
  • Rice, J. A. (2007). Mathematical Statistics and Data Analysis. Cengage Learning.
  • Casella, G., & Berger, R. L. (2002). Statistical Inference. Duxbury.
  • Agresti, A., & Franklin, C. (2017). Statistical Methods for the Social Sciences. Pearson.
  • Blum, W., & Brannick, T. (2006). Psychology: Empirical Research and Evidence-Based Practice. Springer.
  • Stuart, A., & Ord, J. K. (2014). Kendall's Advanced Theory of Statistics. Wiley.
  • DeGroot, M. H., & Schervish, M. J. (2012). Probability and Statistics. Pearson.

Related Assignments