A Certain Statistics Instructor Participates In Triathlons
A Certain Statistics Instructor Participates In Triathlons The Accomp
A certain statistics instructor participates in triathlons. The accompanying table lists times (in minutes and seconds) he recorded while riding a bicycle for five laps through each mile of a 3-mile loop. Use a 0.05 significance level to test the claim that it takes the same time to ride each of the miles. Does one of the miles appear to have a hill?
Determine the null and alternative hypotheses.
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Introduction
The objective of this analysis is to determine whether there are significant differences in the time taken to ride each mile of a 3-mile loop based on the instructor's recorded times. By examining the data statistically, we aim to assess if the riding times are consistent across miles, indicating flat terrain, or if deviations suggest the presence of a hill influencing certain miles.
Data Preparation and Hypotheses
The data collected comprises the times (in minutes and seconds) for five laps through each mile of the loop. To conduct the statistical tests appropriately, all times must be converted into seconds for consistency and ease of analysis.
The null hypothesis (H0) posits that there is no difference in mean riding times across the three miles, suggesting a flat course without hills affecting specific miles:
H0: μ1 = μ2 = μ3
where μ1, μ2, and μ3 are the true mean times for miles 1, 2, and 3, respectively.
The alternative hypothesis (H1) suggests that at least one mile's mean riding time differs from the others, potentially due to elevation changes such as hills:
H1: Not all μi are equal (i.e., at least one μi ≠ another).
This setup reflects a typical one-way ANOVA framework, which tests for differences among group means.
Methodology
A one-way ANOVA is appropriate because it compares the means of more than two groups—in this case, three miles. After converting all lap times into seconds, the mean times per mile are calculated. The ANOVA test examines whether observed differences among means are statistically significant or could have arisen by chance under the assumption of equal true means.
Preliminary analysis involves verifying assumptions: normality of data within each group, independence of observations, and homogeneity of variances. Once these assumptions are satisfied, the ANOVA test proceeds at the 0.05 significance level.
Interpreting Results and Conclusion
If the p-value obtained from the ANOVA is less than 0.05, we reject the null hypothesis, indicating significant differences in riding times among miles. Such a difference would suggest that one of the miles perhaps contains a hill or other elevation change affecting the rider's speed.
Conversely, if the p-value exceeds 0.05, we do not reject the null hypothesis, supporting the claim that ride times are consistent across miles, implying a flat course.
Final Remarks
This analysis provides practical insights into the nature of the course based on riding times. Detecting a significant difference points to terrain variations, which could influence training and race strategies. This approach demonstrates how statistical testing can analyze physical performance data to infer environmental features like hills.
References
- Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics. Sage.
- Moore, D. S., & McCabe, G. P. (2006). Introduction to Statistical Quality Control. Pearson.
- Ott, R. L., & Longnecker, M. (2010). An Introduction to Statistical Research. Cengage Learning.
- Glantz, S. A. (2012). Primer of Biostatistics. McGraw-Hill Education.
- Zar, J. H. (2010). Biostatistical Analysis. Pearson.
- Rumsey, D. J., & Hochster, J. (2004). Statistics for Dummies. Wiley.
- Bickel, P. J., & Doksum, K. A. (2015). Mathematical Statistics: Basic Ideas and Selected Topics. CRC Press.
- Lehmann, E. L., & Romano, J. P. (2005). Testing Statistical Hypotheses. Springer.
- Locker, D., & Cheung, T. (2013). Statistics in Practice. McGraw-Hill.
- Johnson, R. A., & Wichern, D. W. (2007). Applied Multivariate Statistical Analysis. Pearson.