# Detecting Bid-Rigging in Public Procurement. A Cluster Analysis Approach
## 3 Results  
### Testing the Statistical Hypotheses  
The first test was to verify whether there are statistically significant differences between the prices awarded in 2015-2016 and those from 2016-2017, calculated as price/km.  
The statistical assumptions that were tested were as follows:  
\(H_{0}:\mu_{1}=\mu_{2}\) (there are no statistically significant differences between the 2015-2016 and 2016-2017 of the auctions in terms of the actual price paid per kilometer);  
\(H_{a}:\mu_{1}\
eq\mu_{2}\) (there are statistically significant differences between the 2015-2016 and 2016-2017 auctions in terms of the price actually paid per kilometer).  
Testing these statistical assumptions was performed using Student's \(t\)-test. The results of this test are shown in Table 1.  
From this table, it can be noticed that there are significant differences in bid data during the period 2015-2016 to the period 2016-2017 (t\({}_{0.05,42}=-4.67\), \(p\)-value = 0.000). This shows that the price paid per kilometer of removed snow has significantly increased between the two analyzed periods.  
To test from a statistical point of view, the differences between the two periods, regarding the contract price per skimmed and the actual price paid, an analysis was made both for the period 2015-2016 and for 2016-2017.  
The statistical assumptions that were tested were as follows:  
\(H_{0}:\mu_{1}=\mu_{2},\ \mu_{3}=\mu_{4}\) (there are no statistically significant differences between the contract price per kilometer paid for the 2015-2016 or 2016-2017 bids)  
\(H_{a}:\mu_{1}\
eq\mu_{2},\ \mu_{3}\
eq\mu_{4}\) (there are statistically significant differences between the contract price per kilometer paid for the 2015-2016 and 2016-2017 bids)  
Testing of these statistical hypotheses was carried out by Student's \(t\)-test. The results of this test are shown in Table 2.  
The above table indicates that there are statistically significant differences between the contract price and the price actually paid for each kilometer of snow removed, both for the period 2015-2016 (t = 12.75, \(p\)-value = 0.000) and 2016-2017 (t = 15.71, \(p\)-value = 0.000).  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & **Mean** & **Std. Dev.** & **Std. Error Mean** & \multicolumn{4}{c}{**95\% Confidence Interval**} & \multirow{4}{*}{**t**} & \multirow{4}{*}{**Alg.**} \\ \cline{4-9}  & & & & & & **Lower** & & **Upper** & \\ \hline
2015–2016 vs. & \(-\)2839.8 & 3988.25 & 608.2 & \(-\)4067.26 & \(-\)1612.5 & \(-\)4.67 & 42 & 0 \\ \hline \hline \end{tabular} Source: own processing using Statistical Package for the Social Sciences (SPSS) 23 software.  
\end{table}
Table 1: Paired test.  
\begin{table}
\begin{tabular}{c c c c c c c c c} \hline \hline  & **Mean** & **Std. Dev.** & **Std. Error Mean** & \multicolumn{4}{c}{**95\% Confidence Interval**} & \multirow{4}{*}{**t**} & \multirow{4}{*}{**Alg.**} \\ \cline{4-9}  & & & & & & **Lower** & & **Upper** & \\ \hline Estimate\_2015– & & & & & & & & & \\ Effective\_2016 & 16,757.86 & 8621.47 & 1314.76 & 14,104.56 & 19,411.16 & 12.75 & 42 & 0.000 \\ Estimate\_2016– & 10,746.66 & 4838.09 & 684.21 & 9371.69 & 12,121.63 & 15.71 & 49 & 0.000 \\ Effectv\_2017 & & & & & & & & & \\ \hline \hline \end{tabular} Source: own processing using Statistical Package for the Social Sciences (SPSS) 23 software.  
\end{table}
Table 2: Paired sample test.  
It should be noted that for SDN 5, the contract for the period 2016-2017 was divided into smaller batches, and thus, the number of degrees of freedom (No. of contracts awarded--1) differs 2016-2017 (df = 49) compared to 2015 to 2016 (df = 42).