f=5; n=25 # input data alpha=0.05; R=1000 P=.5:R/R # setup list of null parameters pv=pbinom(f,n,P)-.5*dbinom(f,n,P) # get 1-sided p-values plot(P,pv,pch='.', main='P-value plot for all parameters tested', xlab='Parameter used by Ho (P)', ylab='1-tailed mid-P-value') a2=alpha/2; a1=1-a2 abline(h=c(a1,a2),col=c('blue','red')) par(mfrow=c(1,2)) # do next 2 graphs side-by side i=which(pv[-R]>=a1 & pv[-1]<=a1); i1=i+1 CL=P[i1]+(a1-pv[i1])*(P[i1]-P[i])/(pv[i1]-pv[i]) plot(P[(i-2):(i+3)],pv[(i-2):(i+3)], main='CL', xlab='P', ylab='1-tailed mid-P-value') abline(h=a1,col='blue') abline(v=CL,col='green') i=which(pv[-R]>=a2 & pv[-1]<=a2); i1=i+1 CU=P[i1]+(a2-pv[i1])*(P[i1]-P[i])/(pv[i1]-pv[i]) plot(P[(i-2):(i+3)],pv[(i-2):(i+3)], main='CU', xlab='P', ylab='1-tailed mid-P-value') abline(h=a2,col='red') abline(v=CU,col='green') par(mfrow=c(1,1)) # Reset plot default CL # lower conf. limit CU # upper conf. limit CU-CL # interval width

> CL # lower conf. limit [1] 0.07718503 > CU # upper conf. limit [1] 0.3890471 > CU-CL # interval width [1] 0.3118621

Note:
• These values of CL & CU were estimated by arithmetic linear interpolation.

• To obtain a similar (or better) result 'by eye', we could (instead) have entered the instruction locator(2) then pointed to where on each graph we thought the P-value intersected the alpha/2 and 1-alpha/2 lines.

• A simpler way to get a more accurate estimate is to use the using the code given above, but increase the number of parameter values tested from (R=) 1000 to 100000.