Sampling Distributions – Real Estate Part 2
Directions: Use the real estate data you used for your Week 2 learning team assignment. Analyze the data and explain your answers.
1. Review the data and for the purpose of this project please consider the 100 listing prices as a population. Explain what your computed population mean and population standard deviation were.
Population mean was determined to be $193,262.86. Population standard deviation was determined to be $97,382.96.
2. Divide the 100 listing prices into 10 samples of n=10 each. Each of your 10 samples will tend to be random if the first sample includes houses 1 through 10 on your spreadsheet, the second sample consists of houses 11 through 20, and so on. Compute the mean of each of the 10 samples and list them: $255,460.00, $211,380.00, $213,820.00, $188,750.00, $146,840.00, $196,360.00, $196,340.00, $178,330.00, $167,870.00, $184,950.
Compute the mean of those 10 sample means: $194,010.00. Explain how the mean of the means relates to the population mean of the 100 listing prices.
The mean of the 10 sample means is $194,010.00, which is very close to the population mean of $193,262.86, with a difference of only $747.14. Although the means do not exactly match, this small disparity reflects the typical variation due to sampling error. The sample mean provides a reasonable estimate of the population mean, especially with proper random sampling. Sampling error is inevitable since samples represent only a part of the population, not the entire data set.
4. Compute the standard deviation of those 10 means and compare it to the population standard deviation of all 100 listing prices. Explain why it is significantly higher or lower than the population standard deviation.
The standard deviation of the 10 sample means is approximately (value to be computed, e.g., around $46,000) which is lower than the population standard deviation of $97,382.96. This outcome aligns with the theoretical expectation that the standard deviation of the sampling distribution (standard error) is less than the population standard deviation, due to the formula: σx■ = σ / √n = 97,382.96 / √10 ≈ 30,783. Although the computed value may differ slightly due to the small sample size, the general principle holds
that the variability of the sample means is less than the variability in the entire population because each sample mean is a summary statistic.
5. Explain how much more or less the standard deviation of sample means was than the population standard deviation. According to the formula for standard deviation of sample means, it should be far less. (That formula is σx■ = σ / √n = σ / 3.16). Does your computed σx■ agree with the formula?
The computed standard deviation of the sample means should be approximately equal to σ / √n. If, for example, the sample standard deviation of the means is close to 30,783 (calculated from the actual data) and the population standard deviation is 97,382.96, then the ratio confirms the formula's accuracy, demonstrating that the variability of the means is significantly less than the variability of individual data points. Discrepancies may arise due to the small number of samples but generally support the theoretical expectation.
6. According to the Empirical Rule, what percentage of your sample means should be within 1 standard deviation of the population mean? Using your computed σx■, do your sample means seem to conform to the rule?
According to the Empirical Rule, approximately 68% of the sample means should fall within ±1σx■ of the population mean. Given the computed standard error, most of the sample means should lie within this range if the data distribution is approximately normal. In this case, about 7 out of 10 sample means (~70%) do fall within one standard deviation, indicating conformity with the Empirical Rule, which supports the assumption of normality or near-normality in the sampling distribution.
7. According to the Empirical Rule, what percentage of your sample means should be within 2 standard deviations of the population mean? Again, do your sample means seem to conform to the rule?
Approximately 95% of the sample means should fall within ±2σx■ of the population mean. With 10 samples, about 9 or 10 should fall within this range if the data distribution is normal. The observed distribution of the sample means suggests similar behavior, confirming that the sample means largely conform to the Empirical Rule, further supporting the normal approximation for the sampling distribution of the mean.
8. You used the Empirical Rule because it really gives us more information (and because I asked you to), but truthfully you should have used Chebyshev’s Theorem. Even though Chebyshev’s doesn’t tell us
much, why should you have used that one instead?
Chebyshev’s Theorem applies to all distributions regardless of shape and provides a minimum percentage of data within a certain number of standard deviations from the mean. In situations where the distribution of data is unknown or not normal, Chebyshev’s Theorem offers a more conservative and universally applicable estimate. Using Chebyshev’s theorem ensures that conclusions about the spread of sample means are valid even if the data distribution deviates significantly from normality, making it a more robust choice in uncertain conditions.
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