Solved – Confidence interval of ratio estimator

You should take a look at the theory of 'ratio estimators' , there are references to find via google (e.g. http://www.math.montana.edu/~parker/PattersonStats/Ratio.pdf). The idea is that you can compute the mean and the variance of a ratio of random variables and then use the mean and variance to define confidence intervals.

But I think you will need larger samples.

@Matthias Diener

So for your example: $y = (10, 10, 4), bar{y}=8$, $x=(8,8,2), bar{x}=6$

the ratio estimator for your speedup is $r=frac{bar{y}}{bar{x}}=frac{8}{6}=1.333$ and the variance of the estimator is $sigma_r^2=frac{1}{bar{x}^2}frac{s_r^2}{3}$, with $s_r^2=frac{1}{3-1}sum_{i=1}^3 (y_i – rx_i)^2$.

The R-code looks like:

y<-c(10, 10, 4) x<-c(8, 8 , 2)  m.x<-mean(x) m.y<-mean(y)  m.x m.y  r<-m.y/m.x r  s.r.2<-1/(length(x)-1) * sum(( y - r * x ) * ( y - r * x))  variance.r<- 1/m.x^2 * s.r.2/length(x)  stand.deviation.r<-sqrt(variance.r)  cat(paste("estimate for speedup:", round(r, digits=3), "with stand. deviation:", round(stand.deviation.r, digits=3)))