## 3 Results
### Calculation of \(\hat{T}I_{0}\) under the Multi-State Perspective  
As it was mentioned in Section 2, thirteen states were considered in the multi-state perspective, and four samples were manufactured for each state. First, the [PERSON]'s test was performed to assure that no outlier was present at the 0.5% level of significance. Following the recommendation of [40] for the case of dimensional characteristics, it was assumed that the values follow a normal distribution. The [PERSON]'s test was then performed to check the hypothesis of homogeneity between states (Figure 5a) and a 0.510 \(p\)-value was obtained. This result indicated that the null hypothesis cannot be rejected, and it was therefore accepted that the widths of the local intrinsic dispersions can be treated as identical. Assuming equal variances, the [PERSON]'s method was applied to cross-compare the locations of local intrinsic dispersions and determine if they could be considered identical. Thirteen states taken two at a time generated 78 possible combinations. The results of the [PERSON]'s test showed that 61 of those comparisons provided a \(p\)-value lower than the \(\alpha\)-level (0.05), which implied that the null hypothesis shall be rejected, and the locations of the intrinsic intervals were significantly different for those combinations. The remaining 17 comparisons conversely, provided \(p\)-values higher than the \(\alpha\)-level, which means that the locations can be considered equal for those pairs of states involved. Table 3 reflects exclusively the results for those pairs of states whose locations were considered to be equal. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Difference of Levels** & **Difference of Means** & **SE of Difference** & **95\% Confidence Interval** & **\(T\)-Value** & **Adjusted \(p\)-Value** \\ \hline S3—S1 & \(-\)0.00225 & 0.00762 & (\(-\)0.01766; 0.01316) & \(-\)0.30 & 0.769 \\ S6—S1 & \(-\)0.00025 & 0.00762 & (\(-\)0.01566; 0.01516) & \(-\)0.03 & 0.974 \\ S7—S1 & \(-\)0.01475 & 0.00762 & (\(-\)0.00066; 0.03016) & 1.94 & 0.060 \\ S10—S1 & \(-\)0.00400 & 0.00762 & (\(-\)0.01941; 0.01141) & \(-\)0.53 & 0.603 \\ S5—S2 & \(-\)0.01050 & 0.00762 & (\(-\)0.02591; 0.00491) & \(-\)1.38 & 0.176 \\ S7—S2 & \(-\)0.01375 & 0.00762 & (\(-\)0.02916; 0.00166) & \(-\)1.3 & 0.079 \\ S9—S2 & \(-\)0.00275 & 0.00762 & (\(-\)0.01816; 0.01266) & \(-\)0.36 & 0.720 \\ S6—S3 & \(-\)0.00200 & 0.00762 & (\(-\)0.01341; 0.01741) & 0.26 & 0.794 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Results of the [PERSON] individual test for difference of means between states 1. Figure 5: Comparison between states: (**a**) Results of the [PERSON]’s test, including the Bonferroni confidence intervals; (**b**) Boxplot for different states. Despite few states having equal locations according to the \(F\)-test, the overall result was that the production of the test part followed a Type 1 global intrinsic production dispersion according to ISO 22514-8 [40], since the widths of local intrinsic dispersions were equal, but the estimation of the differences in location of local intrinsic dispersions was not null. In this situation, the calculation of the \(\hat{T}I_{0}\) shall be done according to Equation (10). \[\hat{T}I_{0}\ =\ \Delta_{m}+\hat{P}_{m}\big{(}D_{i_{L}}+D_{i_{U}}\big{)} \tag{10}\]  
where \(D_{i_{L}}\) is the half-width of the bottom-slope side of the local intrinsic dispersion, \(D_{i_{U}}\) the top-slope side, and \(\Delta_{m}\) the difference between the higher and the lower locations of the local intrinsic dispersion (Equation (11)). \[\Delta_{m}\ =\ \max\big{(}\overline{x}_{j}\big{)}-\max\big{(}\overline{x}_{j} \big{)} \tag{11}\]  
The extreme locations of the local intrinsic dispersions were obtained for S9 (\(\overline{x}_{j}\) = 40.115 mm) and S13 (\(\overline{x}_{j}\) = 40.222 mm), accordingly, \(\Delta_{m}\ =\ 0.107\) mm. The pooled standard deviation (0.0108 mm) was used to calculate the width of the local intrinsic dispersion and \(D_{i_{L}}\ =\ D_{i_{U}}\ =\ 3\sigma\).