## Appendix A Case Study Project 1: Lengshuibe Auxiliary CSGRD Project
(iii) Output of the \(p\)-value: The function returned both the W statistic and the \(p\)-value, which assessed the data normality. (iv) Results from the Python code: \(\bullet\) Paste margin \(p\)-value: \(p=0.364\). \(\bullet\) Mortar margin \(p\)-value: \(p=0.0589\). \(\bullet\) 14 th-day compression strength \(p\)-value: \(p=0.162\). (v) These \(p\)-values indicate whether the data follow a normal distribution. \(\bullet\) Paste margin: \(p>0.05\), approximately normal. \(\bullet\) Mortar margin: \(p>0.05\), approximately normal. (vi) Conclusion: The paste and mortar margins and 14 th-day compression strength data meet the normality assumption. (f) Homoscedasticity: The Variability of One Variable Should Be Similar Across All Values of the Other Variable Based on the principles of the Breusch-Pagan test, the \(p\)-value from the test (paste and mortar margins and the 14 th-day compressive strength) was greater than 0.05, which suggested that there is no evidence to reject the null hypothesis of homoscedasticity, meaning that the variance of the residuals is constant, and the data meet the assumption of homoscedasticity. (iii) Calculation of the Pearson correlation coefficient and statistical significance using the paste margin and the 14 th-day compressive strength test. The calculation for paste margin and compression strength at 14 days is as follows:* Calculation of the mean, derived from Table 20: \[\bar{\chi}_{paste}=\frac{1}{18}\sum_{i=1}^{18}x_{i}=\frac{0.068+0.072+\cdots+0.09 5}{18}=0.0777\] \[\bar{y}_{144}{=\frac{1}{18}\sum_{i=1}^{18}\dot{y}_{i}}=\frac{1.8+2.6+\cdots+1 5.1}{18}=6.3833\]
* Calculation of the Pearson correlation: \[\alpha=P_{tr}-FA_{sv}\times(\frac{FA_{vp}}{FA_{sv}})\]0
* Calculation of the t-statistic: \[\alpha=P_{tr}-FA_{sv}\times(\frac{FA_{vp}}{FA_{sv}})\]1
* Determination of the \(p\)-value: The \(p\)-value corresponding to the t-statistic of 6.27 with \((18-2)=16\) degrees of freedom is approximately 1.07 \(\times\) 10\({}^{-5}\) from statistical tables. This extremely small \(p\)-value indicates that the correlation between the paste margin and compression strength at 14 days is statistically significant. 4. Calculation of the Pearson correlation coefficient and statistical significance using the mortar margin and the 14 th-day compressive strength test. The calculation for mortar margin and compression strength at 14 days is as follows:
* Calculation of the mean from \[\alpha=P_{tr}-FA_{sv}\times(\frac{FA_{vp}}{FA_{sv}})\]2 \[\alpha=P_{tr}-FA_{sv}\times(\frac{FA_{vp}}{FA_{sv}})\]3 \[\alpha=P_{tr}-FA_{sv}\times(\frac{FA_{vp}}{FA_{sv}})\]4
* Calculation of the Pearson coefficient: \[\alpha=P_{tr}-FA_{sv}\times(\frac{FA_{vp}}{FA_{sv}})\]5
* Calculation of the t-statistic: \[\alpha=P_{tr}-FA_{sv}\times(\frac{FA_{vp}}{FA_{sv}})\]6
* Determination of the \(p\)-value: The \(p\)-value corresponding to the t-statistic of \(-2.59\) with \((18-2)=16\) degrees of freedom is approximately 1.96 \(\times\) 10\({}^{-2}\) from statistical tables. This extremely small \(p\)-value indicates that the correlation between the mortar margin and compression strength at 14 days is statistically significant. 5. Remarks: The above steps are the same for the other curing periods in both projects (7, 28, 90, and 180 days) and for both paste and mortar margins. * _[PERSON] et al. (2012)_ [PERSON]; [PERSON]; [PERSON]; [PERSON] Optimization of Concrete by Minimizing Void Volume in Aggregate Mixture System. _J. Adv. Sci. Eng. Res._**2012**, \(2\), 208-222. * [PERSON] (2018) [PERSON]; [PERSON] The permeable character of CSG Dams and their seepage fields. _Complexity_**2018**. [CrossRef]
* _ASTM C_ 33/C33M; Standard Specification for Concrete Aggregates. ASTM: West Consohocken, PA, USA, 2023. * _ACI_ 211.1; Standard Practice for Selecting Proportions for Normal, Heavyweight, and Mass Concrete (ACI 211.1-22). ACI: Detroit, MI, USA, 2022. * _ASTM C150/C150M;_ Standard Specification for Portland Cement. ASTM: West Consohocken, PA, USA, 2022.