## 7 Practical Application of the CNM Precision Validity Score  
To show the practical applicability of the CNM Precision Validity Score, we first discuss the score for different sample sizes. For our data set, the mean downlink throughput is 23.83 Mbit/s, having a standard deviation of 19.56 Mbit/s and a maximum of 167.95 Mbit/s. To precisely quantify the effect for this data set, in addition to the mean value of the sample, the standard error of mean, the confidence interval, the required minimum number of samples \(n_{\text{abs. }}^{min}(\delta^{*})\), resp. \(n_{\text{rel. }}^{min}(\gamma^{*})\), and the corresponding \(Val.\ Score_{\text{prec. }}\) are evaluated for a confidence level of 95% (\(\alpha=0.05\)) of exemplary sample sizes from 10-10,000,000 measurements of the data set with Studentized \(t\) Intervals in Table 3.  
If a precision of \(\delta^{*}=1\) Mbit/s is desired, the table shows how many measurements are needed to fulfill this precision: For \(n_{\text{abs. }}^{min}(\delta^{*})\geq 1491\Rightarrow z_{0.025}\cdot\frac{s}{ \sqrt{m}}\leq 0.99\) and thus, the precision is higher than 1 Mbit/s. The validity score shows the number of measurements made as a percentage of the measurements required for the desired precision. Once the precision was reached, the validity score is '\(\geq 100\%\)'. If, instead, you prefer to tolerate at most a relative error of \(\gamma^{*}=1\%\), the following condition must hold: \(\frac{z_{0.025}\cdot\delta/\sqrt{m}}{\overline{y}}\leq\frac{0.01}{1+0.01}=0.0099\). In our example, this condition is fulfilled for \(n_{\text{rel. }}^{min}(\gamma^{*})\geq 26\),576. Thus, in this case, a sample with 26,576 measurements would lead to a high accuracy of at most 1% inaccuracy relative to the exact mean value when evaluating the mean. \begin{table}
\begin{tabular}{r r r r r r r r} \hline \hline \begin{tabular}{c} **Sample** \\ **Size** \(m\)** \\ \end{tabular} & \(\overline{y}_{S}\) & \(\frac{s}{\sqrt{m}}\) & \(\boldsymbol{Cl_{0.05}}\) & \(\boldsymbol{z_{0.025}}\frac{s}{\sqrt{m}}\) & \(\boldsymbol{n_{\text{abs.}}^{min}(\delta^{*})}\) & \
\begin{tabular}{c} **Val. Score\({}_{\text{prec.}}\)** \\ **(abs.: \(\delta^{*}\))** \\ \end{tabular} & \(\boldsymbol{n_{\text{rel.}}^{min}(\gamma^{*})}\) &
\begin{tabular}{c} **Val. Score\({}_{\text{prec.}}\)** \\ **(relative: \(\gamma^{*}\))** \\ \end{tabular} \\ \hline
10 & 33.54 & 9.26 & [12.59;54.49] & 20.96 & 4390.61 & 0.002 (0.2\%) & 39,823.66 & \textless{}0.001 (<0.1\%) \\
100 & 21.20 & 1.84 & [17.54;24.86] & 3.66 & 1338.36 & 0.074 (7.4\%) & 30,368.21 & 0.003 (0.3\%) \\
1000 & 23.89 & 0.62 & [22.67;25.11] & 1.22 & 1473.70 & 0.679 (67.9\%) & 26,348.74 & 0.038 (3.8\%) \\
1491 & 23.51 & 0.51 & [22.52;24.50] & (\(\delta^{*}=1\) Mbit/s) 0.99 & 1489.89 & \textgreater{}100\% & 27,491.78 & 0.054 (5.4\%) \\
10,000 & 23.51 & 0.19 & [23.13;23.89] & 0.38 & 1462.00 & \textgreater{}100\% & 26 976.23 & 0.371 (37.1\%) \\
26,576 & 23.57 & 0.12 & [23.34;23.80] & (\(\gamma^{*}=1\%\)) 0.23 & 1447.80 & \textgreater{}100\% & 26,575.75 & \textgreater{}100\% \\
100,000 & 23.82 & 0.06 & [23.70;23.94] & 0.12 & 1473.98 & \textgreater{}100\% & 26,495.38 & \textgreater{}100\% \\
1,000,000 & 23.85 & 0.02 & [23.81;23.89] & 0.04 & 1473.52 & \textgreater{}100\% & 26,431.12 & \textgreater{}100\% \\
100,000 & 23.83 & 0.01 & [23.82;23.84] & 0.01 & 1470.10 & \textgreater{}100\% & 26,405.05 & \textgreater{}100\% \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Precision of different sample sizes. _Validity Score\({}_{\text{prec. }}\) for \(\delta^{*}=1\) Mbit/s_ or \(\gamma^{*}=1\%\) with Studentized \(t\) Intervals. The values of \(n_{abs. }^{min}(\delta^{*})\) and \(n_{rel. }^{min}(\gamma^{*})\) differ. This is especially the case with a small sample size compared to that of a larger sample size. As described in Section 6.2, this is due to the estimation of the standard deviation from the different samples for the absolute case and to the estimation of the standard deviation and the mean value in the relative case.