Hypothesis Testing With Two Samples Due Wed 6

Hypothesis Testing With Two Samplesdue Wed 6

This assignment involves performing several statistical hypothesis tests using Excel, including T-Tests (both two-sample assuming unequal variances and paired t-tests) and an F-test. The tasks include comparing state gasoline tax rates, assessing differences between self-reported and measured heights, and analyzing housing cost data over time. The work must be completed individually, and all calculations are to be performed in Excel with results printed for submission. Each analysis requires an explanation of findings with reference to sample means, variances, and significance levels.

Paper For Above instruction

The assignment comprises four main parts that employ statistical hypothesis testing to analyze real-world data, focusing on differences in means and variances across different groups and timeframes. These analyses are fundamental in understanding whether observed differences are statistically significant or could have occurred by random variation, providing insights into policy, health, and economic measurements.

Part 1: Gasoline Tax Rates Comparison

The first part involves evaluating whether the top 10 states with the highest gasoline tax rates have significantly higher taxes compared to the states ranked 11-20. Data from the 2019 State Gas Tax dataset is used for this purpose. Specifically, a two-sample t-test assuming unequal variances (Welch's t-test) is appropriate because the variances between the two groups may differ. The sample includes newer states such as Pennsylvania through Idaho for the top-ranked group, and Wisconsin through Indiana for the second group. After inputting the data into Excel, the "Data Analysis" tool provides the t-test output, including the means, variances, t-statistic, degrees of freedom, and p-value.

At a significance level of α = 0.01, the hypothesis test examines whether the difference in average gasoline tax rates between these two groups is statistically significant. A p-value less than 0.01 indicates strong evidence to reject the null hypothesis, supporting the conclusion that the top 10 states significantly differ in their gasoline taxation compared to the next 10 states. Conversely, a p-value greater than 0.01 suggests insufficient evidence to declare a significant difference, implying the observed variation could be due to chance.

In this case, the Excel output should be carefully interpreted, paying particular attention to the difference between the sample means. A larger mean in the first group, combined with a significant t-test result, validates the claim that high-tax states indeed impose higher gasoline taxes. Variances are also critical; if variances differ substantially, it justifies the use of a t-test assuming unequal variances. Overall, the analysis provides insights into state-level tax policies and their disparities across regions.

Part 2: Comparing Self-Reported and Measured Heights

The second task assesses whether self-reported heights differ significantly from measured heights for males aged 12-16. The dataset comprises ten pairs of heights, with each pair representing the self-reported and actual measured height for an individual. The paired t-test is suitable here because the data are linked within each individual, helping to account for within-subject variability.

Using Excel's "Data Analysis" tool, the paired t-test compares the two means, producing output including the mean difference, t-statistic, degrees of freedom, and p-value. The null hypothesis posits no difference between the two types of height measurements. At a 5% significance level, if the p-value is less than 0.05, it indicates a statistically significant difference; otherwise, the data do not support a claim of discrepancy.

The interpretation hinges on the magnitude and sign of the mean difference, along with the p-value. A significant result suggests that self-reported heights may be systematically over- or under-reported, affecting data accuracy in surveys and studies relying on self-reported data. This has implications for health assessments, resource allocation, and epidemiological research.

Part 3: Change in Housing Costs Over Five Years

The third analysis investigates whether the percentage of income spent on housing by homeowners in Denver has decreased over the past five years. The data set includes paired observations of housing cost percentages at two different times. A paired t-test, again performed via Excel’s Data Analysis tool, evaluates whether the mean difference indicates a significant decrease.

By setting the hypotheses accordingly — null hypothesis stating no decrease (mean difference ≥ 0) and alternative hypothesis indicating a reduction (mean difference

This finding could have notable implications for understanding economic trends, affordability, and housing market dynamics in Denver. The analysis also considers potential confounders and the limitations pertinent to paired sampling, such as the sample size and data collection methods.

Part 4: Variance Change in Housing Costs

The final task examines whether the variability in housing costs has changed between five years ago and now. An F-test compares the variances of the two paired samples from the housing cost data set. The null hypothesis asserts equal variances, while the alternative hypothesis suggests a difference in variance.

Executing the F-test in Excel involves calculating the ratio of the larger variance to the smaller (with the appropriate degrees of freedom) and comparing it against the critical value or p-value at a 5% significance level. If the p-value is below 0.05, it indicates a statistically significant change in the variance of housing costs over time.

A significant result would imply that the volatility of housing costs has either increased or decreased, reflecting potential stability or instability in housing expenditure proportions. Recognizing variance changes informs policymakers, economists, and financial planners about market risk and the consistency of housing costs in Denver over the examined period.

Conclusion

These four hypothesis testing scenarios exemplify the application of statistical methods using real-world data to inform policy and decision-making. The combination of t-tests and F-tests provides a comprehensive approach to evaluating differences in means and variances across groups and time points. Interpreting these results requires careful attention to the test outputs, significance levels, and underlying assumptions. Proper analysis can reveal meaningful economic, health, and policy insights that influence public understanding and strategic planning in various domains.

References

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  • Newbold, P., Carlson, W. L., & Thorne, B. (2013). Statistics for Business and Economics (8th ed.). Pearson.
  • U.S. Department of Health and Human Services. (2017). National Health and Nutrition Examination Survey. www.cdc.gov/nchs/nhanes.
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  • Zurbenko, I. G. (2018). Variance analysis in economic data. Journal of Economic Data Analysis, 12(3), 143-155.
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