CombineEstimates.nb

CombineEstimates[es_] := Module[{result, ds, var, tvals, newe, t},

   (*Gets the standard deviation for a confidence interval `e`.*)
   Deviation[e_, 
     confidence_] := (Max[e] - 
       Min[e])/(InverseCDF[NormalDistribution[], confidence]*2);
       
   (*Gets the confidence for an interval given its standard
     deviation.*)
   Confidence[e_, dev_] := 
    CDF[NormalDistribution[], ((Max[e] - Min[e])/dev)/2];
   ds = Map[Deviation @@ # &, es];
   
   (*Find the minimum variation `var` and
     the t-values that yield it.*)
   {var, tvals} = 
    Minimize[{Plus @@ Table[t[i]^2*ds[[i]]^2, {i, 1, Length[ds]}], 
      Plus @@ Table[t[i], {i, 1, Length[ds]}] == 1}, 
     Table[t[i], {i, 1, Length[ds]}]];
     
   (*The combined estimate*)
   newe = 
    Fold[#1 + #2[[1]]*#2[[2]] &, 0, 
     Transpose[{es[[All, 1]], Values[tvals]}]];
     
   result = {newe, Confidence[newe, Sqrt[var]]}
   ];