\relax \@writefile{toc}{\contentsline {section}{\numberline {1}Introduction}{1}} \newlabel{sec:intro}{{1}{1}} \@writefile{toc}{\contentsline {section}{\numberline {2}$p$-values and equivalent Gaussian significance}{1}} \newlabel{sec:pval}{{2}{1}} \newlabel{eq:sigmage}{{1}{2}} \newlabel{eq:sigmage2}{{2}{2}} \newlabel{eq:sigmageerf}{{3}{2}} \@writefile{lof}{\contentsline {figure}{\numberline {1}{\ignorespaces {\relax \fontsize {9}{11}\selectfont \abovedisplayskip 8\p@ plus2\p@ minus4\p@ \abovedisplayshortskip \z@ plus\p@ \belowdisplayshortskip 4\p@ plus2\p@ minus2\p@ \def \leftmargin \leftmargini \parsep 4.5\p@ plus2\p@ minus\p@ \topsep 9\p@ plus3\p@ minus5\p@ \itemsep 4.5\p@ plus2\p@ minus\p@ {\leftmargin \leftmargini \topsep 4\p@ plus2\p@ minus2\p@ \parsep 2\p@ plus\p@ minus\p@ \itemsep \parsep }\belowdisplayskip \abovedisplayskip Illustration of the correspondence between the significance $Z$ and a $p$-value.} \label {fig:sigmage}}}{2}} \newlabel{fig:sigmage}{{1}{2}} \@writefile{toc}{\contentsline {section}{\numberline {3}A simple counting experiment}{2}} \newlabel{sec:simplecount}{{3}{2}} \newlabel{eq:pvalpoisson}{{4}{3}} \newlabel{eq:pvalpoisson2}{{5}{3}} \@writefile{toc}{\contentsline {section}{\numberline {4}Simple counting experiment using the likelihood ratio}{3}} \newlabel{sec:simplecountlr}{{4}{3}} \newlabel{eq:expectn}{{6}{3}} \newlabel{eq:lambda}{{7}{3}} \newlabel{eq:muhat}{{8}{3}} \citation{Wilks} \citation{Kendall2} \newlabel{eq:qmu}{{10}{4}} \newlabel{eq:q0}{{11}{4}} \newlabel{eq:lrpval}{{12}{4}} \newlabel{eq:pvalintegral}{{13}{4}} \newlabel{eq:lambdamu0}{{14}{4}} \newlabel{eq:fq00}{{15}{5}} \newlabel{eq:udef}{{16}{5}} \newlabel{eq:pdfsqrtq}{{17}{5}} \newlabel{eq:pvalqobs}{{18}{5}} \newlabel{eq:sigqobs}{{19}{5}} \newlabel{eq:zqobsphihalf}{{20}{5}} \newlabel{eq:Zofn}{{21}{5}} \@writefile{toc}{\contentsline {section}{\numberline {5}Sensitivity for a simple counting experiment}{5}} \newlabel{sec:sensitivity}{{5}{5}} \newlabel{eq:ZPoissonMed}{{22}{6}} \newlabel{eq:ZWMed}{{23}{6}} \newlabel{eq:smallsZW}{{24}{6}} \@writefile{toc}{\contentsline {section}{\numberline {6}Case with multiple bins and systematic uncertainties}{6}} \newlabel{sec:multisys}{{6}{6}} \newlabel{eq:meannij}{{25}{6}} \newlabel{eq:sipar}{{26}{6}} \newlabel{eq:bipar}{{27}{6}} \newlabel{eq:meanmi}{{28}{7}} \newlabel{eq:likelihoodbinned}{{29}{7}} \newlabel{eq:logL}{{30}{7}} \newlabel{eq:Lfull}{{31}{7}} \newlabel{eq:lambdapl}{{32}{8}} \newlabel{eq:Zfull}{{33}{8}} \@writefile{toc}{\contentsline {section}{\numberline {7}Simple counting experiment with background uncertainty}{8}} \newlabel{sec:countingbkg}{{7}{8}} \newlabel{eq:expectn2}{{34}{8}} \citation{SigCalc} \citation{potter} \newlabel{eq:bcomponents}{{35}{9}} \newlabel{eq:emi}{{36}{9}} \newlabel{eq:tauimc}{{37}{9}} \newlabel{eq:likelihood}{{38}{9}} \newlabel{eq:lambdamu}{{39}{9}} \@writefile{lot}{\contentsline {table}{\numberline {1}{\ignorespaces }}{10}} \newlabel{tab:bcklumi}{{1}{10}} \@writefile{toc}{\contentsline {section}{\numberline {8}Case of a single background component}{10}} \newlabel{sec:singcomp}{{8}{10}} \newlabel{eq:muhat2}{{40}{10}} \newlabel{eq:bhat}{{41}{10}} \newlabel{eq:bhathat0}{{42}{10}} \newlabel{eq:lnlambdapsi0}{{43}{10}} \newlabel{eq:psi}{{44}{11}} \newlabel{eq:lnLambdaExample}{{45}{11}} \newlabel{eq:ZExample}{{46}{11}} \@writefile{lof}{\contentsline {figure}{\numberline {2}{\ignorespaces {\relax \fontsize {9}{11}\selectfont \abovedisplayskip 8\p@ plus2\p@ minus4\p@ \abovedisplayshortskip \z@ plus\p@ \belowdisplayshortskip 4\p@ plus2\p@ minus2\p@ \def \leftmargin \leftmargini \parsep 4.5\p@ plus2\p@ minus\p@ \topsep 9\p@ plus3\p@ minus5\p@ \itemsep 4.5\p@ plus2\p@ minus\p@ {\leftmargin \leftmargini \topsep 4\p@ plus2\p@ minus2\p@ \parsep 2\p@ plus\p@ minus\p@ \itemsep \parsep }\belowdisplayskip \abovedisplayskip The significance $Z$ as a function of the expected signal $s$ according to several formulae (see text).} \label {fig:sZ}}}{11}} \newlabel{fig:sZ}{{2}{11}} \bibcite{CSCcombo}{1} \bibcite{Cranmer05}{2} \bibcite{Wilks}{3} \bibcite{Kendall2}{4} \bibcite{Asimov}{5} \bibcite{SigCalc}{6} \@writefile{lof}{\contentsline {figure}{\numberline {3}{\ignorespaces {\relax \fontsize {9}{11}\selectfont \abovedisplayskip 8\p@ plus2\p@ minus4\p@ \abovedisplayshortskip \z@ plus\p@ \belowdisplayshortskip 4\p@ plus2\p@ minus2\p@ \def \leftmargin \leftmargini \parsep 4.5\p@ plus2\p@ minus\p@ \topsep 9\p@ plus3\p@ minus5\p@ \itemsep 4.5\p@ plus2\p@ minus\p@ {\leftmargin \leftmargini \topsep 4\p@ plus2\p@ minus2\p@ \parsep 2\p@ plus\p@ minus\p@ \itemsep \parsep }\belowdisplayskip \abovedisplayskip The significance $Z$ as a function of the luminosity ratio $\tau = L_{\rm MC}/L_{\rm data}$ for several values of $b$ and $s$.} \label {fig:tauZ}}}{12}} \newlabel{fig:tauZ}{{3}{12}} \bibcite{potter}{7}