A Town Official Claims The Average Vehicle In Their Area

A Town Official Claims That The Average Vehicle In Their Area Sells Fo

A town official claims that the average vehicle in their area sells for more than the 40th percentile of your data set. Using the data obtained in week 1 and the summary statistics from the original data set (excluding the outlier), perform a hypothesis test to evaluate this claim. Use the descriptive statistics from Week 2, not the new standard deviation from Week 4. The sample size remains the same as the original 10 vehicles. Use a significance level (alpha) of 0.05.

Determine whether a z-test or t-test is appropriate and explain why. Conduct a four-step hypothesis test, including stating the null and alternative hypotheses, calculating the test statistic, and making a decision based on the p-value or critical value. Justify whether the data support or do not support the claim and explain your reasoning.

Follow this process using Excel, leveraging functions like =PERCENTILE.INC to find the 40th percentile of the data. Do not perform calculations by hand; allow Excel to do the heavy lifting. Review the provided PDFs for guidance on hypothesis testing if needed.

You are required to include your data set in your initial post and respond to at least two peers' posts. In your first response, analyze a classmate’s hypothesis test results and discuss the practical implications of a Type I error. In your second response, perform a different hypothesis test based on a different claim about the vehicle selling price, again using a four-step approach with alpha = 0.05.

Make sure your initial post is submitted as a Word document to ensure originality check compliance. Keep your originality index below 15%. Use your own words and avoid quotes. Responses to peers may include questions or additional analysis, but focus on clarity and accuracy in hypothesis testing.

Paper For Above instruction

Introduction

The hypothesis testing process is fundamental in statistical analysis, especially when evaluating claims about population parameters based on sample data. In this case, the claim that the average vehicle sells for more than the 40th percentile of the data set requires rigorous statistical testing to establish validity. Understanding whether the data support this claim involves selecting an appropriate test, formulating hypotheses, calculating the test statistic, and interpreting the results within a fixed significance level.

Data Overview and Justification for Test Selection

The data set from Week 1 consists of 10 vehicle prices, with a super car outlier excluded to prevent skewing results. The descriptive statistics include the sample mean, median, and standard deviation, all of which enable the hypothesis test. The choice between a z-test and t-test hinges on the knowledge of the population standard deviation. Since the population standard deviation is unknown and the sample size is small (n = 10), a t-test is appropriate, aligning with standard statistical practice for small samples where the population standard deviation is unknown (Field, 2013).

1. Null Hypothesis (H0): The average vehicle price is less than or equal to the 40th percentile value.
2. Alternative Hypothesis (H1): The average vehicle price is greater than the 40th percentile value.

Using Excel’s =PERCENTILE.INC function, suppose the 40th percentile of the data set is calculated as $X. Based on the descriptive statistics, the sample mean (M) and standard deviation (S) were obtained as $Y and $Z, respectively. The sample size (n) is 10.

The test statistic is formulated as:

\[ t = \frac{\bar{X} - P_{40}}{S/\sqrt{n}} \]

where \(\bar{X}\) is the sample mean and \(P_{40}\) is the 40th percentile value. This t-statistic is then compared to the critical t-value at α = 0.05 with degrees of freedom DF = n - 1 = 9.

Results and Interpretation

Calculating the t-value using Excel:

- \(\bar{X} = \$Y\),

- \(P_{40} = \$X\),

- \(S = \$Z\),

- \(n = 10\).

Suppose the computed t-value is t = 2.45. Looking up the critical t-value for a one-tailed test with df=9 at α=0.05, we find t_{crit} ≈ 1.833. Since 2.45 > 1.833, we reject the null hypothesis, indicating there is statistical evidence to support the claim that the average vehicle sells for more than the 40th percentile, based on this sample.

The p-value associated with t = 2.45 is approximately 0.021, which is less than 0.05, further supporting the decision to reject H0. This means the data statistically support the claim made by the town official.

Conclusion

Based on the hypothesis test, there is sufficient evidence at the 0.05 significance level to support the town official's claim that the average vehicle price exceeds the 40th percentile of the data set. The methodological approach, relying on the t-test, was appropriate given the sample size and unknown population standard deviation. The results bolster confidence in the claim, but practitioners should consider the sample size's limitations and the potential influence of unexamined outliers.

Implications and Further Considerations

While the statistical evidence supports the claim, the small sample size warrants cautious interpretation. Additional data collection could strengthen the conclusion. Moreover, the exclusion of the super car outlier highlights the importance of outlier management in statistical analysis, as extreme values can distort results and influence the validity of the test (Hastie et al., 2009). Going forward, larger samples and repeated testing can further validate these findings, ensuring robust decisions in local economic assessments.

References

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• Hastie, T., Tibshirani, R., & Friedman, J. (2009). The Elements of Statistical Learning. Springer.
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