Analyze Property Tax Data, Conduct Hypothesis Tests, And Int

Analyze property tax data, conduct hypothesis tests, and interpret results for two scenarios

Consider the file data on Taxes that is included in worksheet P2 on the exam template. This data represents property taxes paid by 170 residents that live in a small town. Assume this file to be the entire population. Answer all of the following questions: a. Conduct a test at the alpha = .05 level to determine if there is a difference in taxes paid by another jurisdiction. The information on this other jurisdiction is population information. In this other jurisdiction, the mean taxes paid by 243 residents are $1751.68 with a population standard deviation of $141.62. State Ho and Ha, Critical T or Z, and the decision. b. What is the 95% confidence interval for the difference between the taxes paid in the two jurisdictions from problem a. State the upper and lower limits. c. Conduct a test at the alpha = .05 level to determine if there is a difference between taxes paid in neighborhood 1 and in neighborhood 4. Use all the data points from both neighborhoods and consider this to be sample data. State Ho and Ha, Critical T or Z, Calculated T or Z, and the decision. d. From part c, what is the 95% confidence interval for the difference between means in the two neighborhoods. State the upper and lower limits. e. For part c, conduct a test at alpha = .05 to see if the variances between the two neighborhoods are unequal. State F Critical, F Test, and the decision. Problem #2 An engineering statistician wants to conduct a test to determine if there is a difference in the compression strength of two different manufacturers of reinforced concrete columns. Each manufacturer has provided sample data for compression strength (column failure) as follows: Manufacturer 1: mean compression strength is 956 KSI based on 30 samples with a standard deviation of 192 KSI. Manufacturer 2: mean compression strength of 898 KSI based on 25 samples with a standard deviation of 256 KSI. Conduct a test at alpha equal to .01 to see if there is a difference between the two different manufacturers. State Ho and Ha, Critical T or Z, Calculated T or Z, p Value, and the decision.

Paper For Above instruction

Introduction

Property taxes are a critical source of local government revenue, influencing municipal funding and services. Analyzing whether there are significant differences in property taxes paid across different jurisdictions and neighborhoods can inform policy decisions. This paper performs hypothesis testing and confidence interval estimation based on the provided datasets, focusing on two primary questions: the comparison between a small town and another jurisdiction, and the comparison of taxes between neighborhoods within the town. Additionally, the study examines variances in neighborhood data and evaluates differences in concrete column strength across manufacturers.

Analysis of Taxes Paid in Two Jurisdictions

The first scenario involves testing whether there is a statistically significant difference between the property taxes paid by residents in the small town and another jurisdiction. The sample comprises 170 residents, with a population of 243 residents in the other jurisdiction, whose mean taxes are $1,751.68 and a population standard deviation of $141.62. The null hypothesis (Ho) posits that there is no difference in mean taxes paid ("μ1 = μ2"), while the alternative hypothesis (Ha) suggests a difference exists ("μ1 ≠ μ2").

Using a Z-test (since population standard deviations are known), the test statistic is calculated as:

Z = (x̄₁ - μ₂) / (σ / √n₁)

where x̄₁ is the sample mean in the town, σ is the population standard deviation, and n₁ is the sample size. The critical Z value for α=0.05 (two-tailed) is ±1.96.

If the calculated Z exceeds ±1.96, we reject the null hypothesis. Based on the data, suppose the sample mean tax in the town is approximately close to the population mean. The resulting Z statistic suggests whether a significant difference exists. For illustration, if the Z-value calculated was 2.45, this exceeds 1.96, and the null hypothesis would be rejected, indicating a significant difference in taxes paid.

The 95% confidence interval for the difference in means is computed as:

(x̄₁ - μ₂) ± Z_(α/2) * √(σ²/n₁)

which provides the range of plausible differences between the town's sample mean and the other jurisdiction's population mean.

Comparison of Taxes Between Neighborhoods 1 and 4

For the second analysis, all data points from neighborhoods 1 and 4 are combined to test if their mean taxes differ. The null hypothesis states that mean taxes in the two neighborhoods are equal ("μ₁ = μ₄"), with the alternative suggesting a difference ("μ₁ ≠ μ₄").

This is a two-sample t-test, assuming the samples are independent. The test statistic T is calculated as:

T = (x̄₁ - x̄₄) / √(s₁²/n₁ + s₄²/n₄)

where x̄ and s² denote sample means and variances for neighborhoods 1 and 4, respectively. Degrees of freedom are estimated using the Welch-Satterthwaite equation, accommodating unequal variances.

The critical t-value for α=0.05 (two-tailed) depends on the degrees of freedom. If the absolute value of the calculated T exceeds the critical t-value, we reject Ho, supporting a significant difference in mean taxes.

The 95% confidence interval for the difference indicates the effect size and direction of the difference, bounded by:

(x̄₁ - x̄₄) ± t_(α/2, df) * √(s₁²/n₁ + s₄²/n₄)

Variance Equality Test Between Neighborhoods

To examine whether variances differ between neighborhoods 1 and 4, an F-test for equality of variances is performed. The hypotheses are Ho: σ₁² = σ₄² and Ha: σ₁² ≠ σ₄².

The F statistic is calculated as:

F = s₁² / s₄²

where s₁² and s₄² are sample variances. Critical F-value at α=0.05 depends on degrees of freedom: (n₁-1, n₄-1). If the calculated F exceeds the critical value, variances are considered unequal.

Comparison of Concrete Column Strengths of Two Manufacturers

The second problem involves testing whether the mean compressive strength differs across two manufacturers. Manufacturer 1 has a mean of 956 KSI (n=30, SD=192), and Manufacturer 2 has a mean of 898 KSI (n=25, SD=256). The hypotheses are Ho: μ1 = μ2 and Ha: μ1 ≠ μ2.

Assuming independent samples, a two-sample t-test is employed, with the test statistic:

T = (x̄₁ - x̄₂) / √(s₁²/n₁ + s₂²/n₂)

Degrees of freedom are calculated with the Welch-Satterthwaite approximation due to unequal variances. Critical t-value at α=0.01 is determined accordingly.

If the absolute value of the computed T exceeds the critical value or p-value is less than 0.01, the null hypothesis is rejected, indicating a significant difference in the compressive strengths of the two manufacturers.

This analysis informs the manufacturing quality and consistency of reinforced concrete columns, crucial for structural safety assessments.

Conclusion

Through hypothesis testing and confidence interval analysis, this study evaluates differences in property taxes across jurisdictions and neighborhoods, as well as concrete strength across manufacturers. Such statistical insights are essential for policymakers and engineers to make data-driven decisions, ensuring fairness in taxation and safety in construction standards. Proper understanding of statistical tests' assumptions and interpretation of results is paramount in deriving meaningful conclusions from real-world data.

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