## 4 Discussion
### Correlation Analysis: 4 in. vs. 6 in. Cylinders (Curing Method #C2)  
There are 192 observations in the paired dataset for the correlation analysis of the compressive strengths of 4 in. cylinders and 6 in. cylinders cured using only Method #C2. The degrees of freedom of residuals is 192 \(-\) 2 = 190. The degree of freedom of the model is 1. The R-squared value is 0.913. The Adjusted R-squared value is 0.912 for the number of predictors in the model. The F-statistic of 1989 is greater than the critical value of 3.890, which means that the coefficient related to the predictor variable (the compressive strengths of 4 in. cylinders) is non-zero. The Prob(F-statistic) is close to zero, which implies that the regression model is meaningful. Figure 6 shows the fitted line plots when using the compressive strengths of 4 in. cylinders as the x-axis and the compressive strengths of 6 in. cylinders as the y-axis. Figure 6: _Cont._  
Figure 7 shows the residual analyses of the influences of Method #C2 when using the residuals as the x-axis and the probability percentages as the y-axis. Figure 7a,e shows the normal probability plots. The points in Figure 7a form an approximate linear pattern, which indicates that a normal distribution can be used for this dataset. The points in Figure 7e have outliers in the lower left corner. The residual versus fits plots in Figure 7b,f detect non-linearity, unequal error variances, and outliers. The residuals distribute randomly around the zero line, which suggests that the assumption that the relationship is linear is reasonable. Additionally, the residuals of \((y_{actual}-y_{predicted})\) are on the y-axis and have a wider span around the zero line above the 3500 psi fitted values of the estimated compressive strengths of the 6 in. cylinders than the ones in the range of 2000 psi to 3500 psi, which suggests that the variances of the error terms increase when the estimated compressive strengths of the 6 in. cylinders increase. Then, the variances stay the same after the 3500 psi value. A few residuals are different from the basic random pattern of residuals, which suggests that there are outliers. Figure 7c shows the histogram of the residual frequency for the corresponding linear regression. The zero residuals have the most frequencies, and the negative and positive residuals are evenly distributed. The histogram has a normal distribution. In Figure 7g, the zero residuals have the most frequencies, and the positive residuals are more than the negative residuals; hence, the histogram is skewed. Figure 7d,h shows the standardized residuals in the order of the corresponding observations of the corresponding regressions. The residuals fluctuate in a random pattern around the center line of zero. No correlation exists between error terms that are near each other. No ascending or descending trend in the residuals is shown in the correlation among residuals. Table 7 compares the performances of the correlation and regression. The correlations were generated using the compressive strengths of the 4 in. cylinders as the x-axis and the compressive strengths of the 6 in. cylinders as the y-axis. There are 192 observations in the paired dataset. The nonlinear analysis assumes three independent variables (x1, x2, and x3) and one dependent variable. In this case, the df of residuals is 192 \(-\) 3 \(-\) 1 = 188. The degree of freedom of the model is 3. The R-squared value and the Adjusted R-square value are 0.904 and 0.903, respectively, which means that 90.40% of the variation in the y values is accounted for by the x values. The F-statistic of 593.1 is greater than the critical value of 2.6790, which means that the alternate hypothesis holds, and the coefficient related to the predictor variable (the compressive strengths of the 4 in. cylinders) is non-zero. The Prob(F-statistic) value is close to zero, which implies that the regression model is meaningful. Figure 6: Fitted line plots about 4 in. vs. 6 in.