black circle (gray square) is the best fit value for the likelihood ($\chi^2$). *Left panel:* Results for the $\Lambda$CDM model with the SDSS compilation. *Right panel:* Results for the F$w$CDM model with the SDSS compilation. \[fig:SDSS\_beta\_alpha\]](FIG3_a.eps "fig:"){width="49.00000%"} ![68% and 95% confidence level contours in the plane $\beta\times\alpha$. We marginalized over all other parameters with flat prior. The solid (dashed) lines are the results for the likelihood ($\chi^2$) approach. The black circle (gray square) is the best fit value for the likelihood ($\chi^2$). *Left panel:* Results for the $\Lambda$CDM model with the SDSS compilation. *Right panel:* Results for the F$w$CDM model with the SDSS compilation. \[fig:SDSS\_beta\_alpha\]](FIG3_b.eps "fig:"){width="49.00000%"}

In Fig. \[fig:SDSS\_sint\_dist\] we show the probability distributions for $\sigma_{\text{int}}$, given by the likelihood approach, for MLCS2k2 and SALT2 data. The traditional $\chi^2$ approach gives only one value for this parameter without uncertainty and we represent it by the dashed vertical line in the figure. We can see that the discrepancy between the value obtained from the $\chi^2$ approach and the best fit value obtained from the likelihood approach is greater for the MLCS2k2 data. The results are incompatible at more than 99% confidence interval, which does not happen for the SALT2 data. This 