Elected Officials In A California City Are Preparing Theannu

29elected Officials In A California City Are Preparing Theannual Budg

Elected officials in a California city are preparing the annual budget for their community. They would like to estimate how much their constituents living in this city are typically paying each year in real estate taxes. Given that there are over 100,000 homeowners in this city, the officials have decided to sample a representative subset of taxpayers and study their tax payments.

a. What sample size is required to generate a 95% confidence interval for the mean annual real estate tax payment with a half-length of $100? Assume that the best estimate of the population standard deviation s is $535.

b. If a random sample of the size from part a is selected and a 95% confidence interval for the mean is calculated from this sample, will the half-length of the confidence interval be equal to $100? Explain why or why not.

c. Now suppose that the officials want to construct a 95% confidence interval with a half-length of $75. What sample size is required to achieve this objective? Again, assume that the best estimate of the population standard deviation s is $535. Explain the difference between this result and the result from part a.

Paper For Above instruction

The process of estimating population parameters through sampling plays a crucial role in efficient and effective governmental decision-making, particularly when managing city budgets and resource allocations. In this context, understanding how to determine an appropriate sample size for estimating the mean real estate tax payments is vital. This paper explores the statistical methodology for calculating necessary sample sizes, the implications of sampling variability, and the practical considerations involved in constructing confidence intervals with specific desired margins of error.

Introduction

Sample size determination is a foundational aspect of statistical inference, especially in survey-based studies where the goal is to estimate a population parameter with a specified level of confidence and precision. For elected officials in a California city interested in approximating the mean annual real estate taxes paid by homeowners, the critical challenge is defining a sample size that balances accuracy with resource constraints. This analysis utilizes the known population standard deviation to derive these sample sizes, assuming a normal distribution of tax payments.

Calculating Sample Size for a 95% Confidence Interval with Half-Length of $100

The first step is to determine the sample size necessary to construct a confidence interval with a specified margin of error, or half-length, which in this case is $100. Given the confidence level of 95%, the critical z-value from the standard normal distribution is approximately 1.96. The formula for the required sample size (n) when the population standard deviation (σ) is known is:

n = (Z * σ / E)²

where Z = 1.96, σ = 535, and E = 100. Plugging in these values:

n = (1.96 535 / 100)² ≈ (1.96 5.35)² ≈ (10.49)² ≈ 110.22

Since sample size must be an integer, rounding up yields n = 111. Therefore, a sample size of at least 111 taxpayers is needed to estimate the mean real estate tax payment with a 95% confidence level and a half-length of $100. This calculation ensures that the margin of error does not exceed $100, given the population standard deviation.

The Implication of Sample Size on Confidence Interval Half-Length

In practice, if a sample of size 111 is randomly selected and used to compute a confidence interval, the theoretical half-length may not be exactly $100. This is because the confidence interval's half-length is an estimate based on the sample data, and natural sampling variability can cause deviations. Specifically, the actual half-length from the sample depends on the observed sample standard deviation and the particular values within that sample. While the calculated sample size ensures that, on average, the margin of error is no greater than $100, individual samples may produce slightly larger or smaller intervals due to sampling fluctuation. Additionally, the approximation assumes the population standard deviation is known; if it is estimated from the sample, the actual half-length may differ further, especially with smaller samples.

Adjusting Sample Size for a Smaller Margin of Error: $75

Suppose the city officials desire a more precise estimate, with a half-length of $75 at the same confidence level. Using the same formula:

n = (Z σ / E)² = (1.96 535 / 75)²

Calculating:

n = (1.96 535 / 75)² ≈ (1.96 7.13)² ≈ (13.97)² ≈ 195.24

Rounding up, the necessary sample size is n = 196. This larger sample size reflects the inverse relationship between margin of error and the sample size; as the desired precision becomes finer (smaller E), the required sample size increases substantially.

The key difference between this result and that found in part a is the magnitude of the margin of error. Reducing the half-length from $100 to $75 nearly doubles the sample size from 111 to 196, illustrating the trade-off between precision and resource expenditure.

Discussion and Practical Implications

These calculations assume the population standard deviation is accurately known, which permits the use of the normal distribution for sample size estimation. In real-world scenarios, this parameter is often estimated from prior data or pilot studies. The increase in sample size required for smaller margins of error signifies the importance of balancing budget constraints with the need for accuracy. Larger sample sizes reduce variability in the estimate and increase confidence in the results, but they also involve greater resource allocation, time, and costs.

Understanding the mathematical relationships involved allows policymakers to make informed decisions about the necessary sampling effort to achieve desired confidence levels and precision, ultimately leading to more reliable budget forecasts and fiscally responsible planning.

Conclusion

Sample size determination is a critical component of statistical inference that enables elected officials to estimate mean real estate taxes accurately. Using the known population standard deviation and a specified confidence level, officials can calculate the necessary sample sizes for different margins of error. Smaller half-lengths demand larger samples, underscoring the importance of strategic planning in survey design. Recognizing the potential variability in confidence intervals, even with the correct sample size, ensures that policymakers interpret their estimates within the appropriate context, facilitating transparent and reliable budget planning.

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