CME 281 Computational Methods Hypothesis Testing

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Analyze a series of computational methods problems, including hypothesis testing, curve fitting, data analysis, and goodness of fit, based on provided data and scenarios. Your task involves performing statistical tests, regression analyses, and data modeling to interpret the data accurately.

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The first problem involves hypothesis testing to determine whether competitors in a game show, Wipeout, are lasting longer this season compared to previous seasons. The previous average event time was 17 minutes, with an unknown standard deviation, and data from the current season shows an average of 21 minutes for 5 competitors with a standard deviation of 2.2 minutes. To test whether there is a significant increase in competitor endurance, a t-test for the mean can be applied at a 0.01 significance level. The hypotheses are: H₀: μ = 17 (no increase) against H₁: μ > 17 (longer duration). Calculating the t-statistic involves the sample mean, the population mean, the sample standard deviation, and the sample size. Since the population standard deviation is unknown and the sample size is small, the t-distribution provides an appropriate model, and the critical value from the t-table indicates the threshold for rejection of H₀. Based on the calculations, if the t-statistic exceeds the critical t-value, the conclusion is that competitors are lasting longer, supporting the hypothesis that the game show's events are becoming easier or more advantageous for participants.

The second problem assesses whether a coffee-filling machine is underfilling jars. The industry standard target is 16 ounces, but a sample of 35 jars shows a mean of 15.8 ounces with a sample standard deviation of 0.6 ounces. The hypothesis test involves H₀: μ = 16 ounces against H₁: μ

For the curve fitting analysis, data points are to be modeled using Excel. Various fits such as linear, polynomial, or exponential are tested to find the best reflection of the data pattern. The correlation coefficients (R²) and residuals are analyzed to assess accuracy. A higher R² indicates a better fit, but residual patterns can reveal the suitability of the model. The most appropriate model should balance statistical goodness-of-fit with interpretability and the pattern of residuals. The review concludes with a discussion of which model best captures the data trend.

The empirical analysis of variable relationships in a stirred tank reactor involves comparing measured data such as temperature, concentration, inflow, and volume variations. Without first-principles models, pairwise comparisons help identify significant correlations. Calculating covariance and correlation coefficients informs about the strength and nature of relationships among variables. The analysis provides insight into potential dependencies, which can inform future modeling efforts or process optimization.

The last segment involves modeling the population dynamics of squirrels engaging with food over time. Data collected from multiple trials is plotted, and trendlines are fitted using Excel's trendline functions. Various models, including linear, polynomial, and exponential, are tested for their fit quality, focusing on correlation coefficient thresholds (e.g., R² > 0.96). Additionally, a custom regression model is proposed: N(t) = A e^{Bt} + C, where A, B, and C are parameters optimized via Excel’s solver. The fitted model is then plotted alongside the data, and its correlation coefficient is computed to compare with other fits. This process helps identify the most accurate mathematical description of the squirrel activity trend over time.

References

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  • O’Neill, R. P. (2011). Practical Regression and Anova using R. SAGE Publications.
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  • Bevington, P. R., & Robinson, D. K. (2013). Data Reduction and Error Analysis for the Physical Sciences. McGraw-Hill Education.

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