Accesos a memoria: tamaño y alineamiento

Because the the data recorded in 2001 started from September to December and the means and variances of year 2001 were relatively lower than the others, as seen in Table 3.4, the equality of the quarterly loss rates were tested to see any seasonal effects existed. The small p − value in the Table 3.16 indicated that significant differences were among the quarterly gasoline loss rates and there was seasonal effect on gasoline loss rates. The large p − value of diesel data indicated that there was no detectable differences among the quarterly diesel loss rates and hence no seasonal effects on the diesel loss rates.

Table 3.16: Chi-square test on quarterly data

Results µ c(µ) p−value

Gasoline 0.3150 27.2049 3.5343e-07

3.2.5 Monthly Results

Detecting any surges of fuel loss rates in certain months and finding out the difference of these months would help determine the causes of fuel loss. In the Table 3.17, the small p− values of gasoline and diesel indicated that big differ- ence existed among monthly loss rates. The much smaller p− values of gasoline indicated that the significance for gasoline data was more extreme than that of diesel’s. The variations among the gasoline loss rates should be relatively larger than that of diesel’s if measured at certain significance level.

Table 3.17: Chi-square test on monthly data

Results µ c(µ) p−value

Gasoline 0.3146 34.3155 0.0003

Diesel -0.1705 25.0979 0.0088

3.2.6 Sitecode Results

Determine whether all the gas stations were experiencing same losses would help detect any specific gas station that had significant loss compared with others.

The performed χ2 _{test results were illustrated in the Equation 3.3 for gasoline}

and Equation 3.4 for diesel. The results showed that the both of them were extremely significant, and there were no doubt that losses of each gas station were statistically different. By looking at the box plots in Figure 3.10 and Figure 3.11 and the means of the fuel loss rates, it was found that, the gas stations 101, 102, 108, 160 and 182 had the highest positive gasoline loss rates, while gas stations 146,177 and 537 had the most negative diesel loss rates.

ˆ

ˆ

µ = −0.0526, c(µ) = 1310.876, p − value = 0 (3.4)

3.3

Multiple Comparisons

After testing the equality of data groups, it is interesting to continue testing the mutual differences of group members that had been identified as different. We used the Tukey Method to perform the analysis [9]. The algorithm of this method

was illustrated in the Equation 3.5. The t statistic to test τi = τj vs. τi 6= τj was

tij = mj− mi ˆ σq 1 nj + 1 ni (3.5)

where ˆσ =√MSE. By the Tukey Method, ”τidifferents from τj” if |tij| >_√1

2qk,N −k,α

where qk,N −k,αwas the upper α quantile of the Studentized range distribution,with

parameter k and N − k degree of freedom.

3.3.1 By Region

At 10% significance level, if |tij| > 3.1143, τi differs from τj. Comparing with the

numbers in Table 3.18, we concluded that, the pairs of LA and SB, LA and SD, SB and OC, OC and SD were significantly different, while the pairs of LA and OC, SB and SD were close.

Table 3.18: Tukey test on regional data

LA vs. SB LA vs. OC LA vs. SD SB vs. OC SB vs. SD OC vs. SD

3.3.2 By Year

Since the χ2 _{tests on diesel yearly loss rates indicated there were not much de-}

tectable difference among the yearly data, we applied the Tukey test only to the gasoline data. The Tukey results were as presented in the Table 3.19, where 1 stands for 2001, 2 stands for 2002 and so on so forth. The loss rates of 2004 and 2005 were close, and the loss rates of 2003, 2006 and 2007 were close. The loss rates of 2001, 2002 and 2008 were significantly different from any of the other years respectively. Combining this statement with the numbers in Table 3.4, it can told that the gasoline loss rates increased from the year 2001 to 2003, and peaked around the years 2004 and 2005. Afterwards, it dropped during 2006, of which the gasoline loss rates were close to that of the year 2003. After 2007, the level of gasoline loss rates only went down little in the 2008.

Table 3.19: Tukey test on yearly data

1 vs. 2 1 vs. 3 1 vs. 4 1 vs. 5 1 vs. 6 1 vs. 7 1 vs.8 5.5984 10.3918 13.8015 12.6674 9.9717 10.1019 7.4340 2 vs. 3 2 vs. 4 2 vs. 5 2 vs. 6 2 vs. 7 2 vs. 8 6.3090 10.8770 9.3494 5.6887 5.8458 2.6129 3 vs. 4 3 vs.5 3 vs. 8 4 vs.6 4 vs.7 4 vs.8 4.5768 2.9603 -3.2706 -5.6379 -5.6274 -7.51010 5 vs.6 5 vs.7 5 vs.8 6 vs.8 7 vs.8 -3.9685 -3.9349 -6.0494 -2.5812 -2.6894 3.3.3 By Quarter

Similarly, since the χ2 _{tests on diesel quarterly loss rates indicated there were}

gasoline data. The results listed in the Table 3.20 showed that, the quarterly gasoline losses were mutually close except the pairs of quarter 1 and quarter 3, and quarter 2 and quarter 3. Combining these statement with the figures in Table 3.6, it can be concluded that gasoline loss rates were stable during the 1st and 2nd quarters. It increased significantly in the 3rd quarter and dropped a little in the 4th quarter and went back to the level that was close to that of the 2nd quarter.

Table 3.20: Tukey test on quarterly data quarter 1 vs. quarter 3 quarter 2 vs. quarter 3

4.9150 3.3493

3.3.4 By Month

Since both the p values of gasoline and diesel data were small by the χ2 _test,

Tukey tests were performed on gasoline monthly data and diesel monthly data respectively. The test results for gasoline monthly data were shown in Table 3.21. Combining the results in Table 3.21 with the numbers in Table 3.8, it can be concluded that the gasoline loss rates were stable from January to June, increased in July and August and then dropped little a bit in September and stayed stable afterwards. The results of Tukey test on diesel data showed there was no one pair was significantly different.

Table 3.21: Tukey test on monthly data 1 vs. 8 3 vs. 7 3 vs. 8

3.3.5 By Sitecode

As concluded in Equation 3.3 and Equation 3.4, the fuel loss rates varied signifi- cantly among different gas stations. Tukey results showed that, at the significant level of 1%, huge difference of gasoline loss rates existed among 2220 pairs of gas stations. Only the pair of gas station 160 and 177 was significantly different in diesel loss rates at the significant level of 5%.