> # Oskar Hagberg 2012-02-15
> # Question
> #
> # Is there a difference in mean performance between the Karlsruhe and Lehigh
> # Method in the accuracy of preidicting sheer strength?
> # 
> # Experiment to answer the question
> # 
> # Nine giders were tested both with the K and the L method. The results were -
> # expressed the ratio between predicted and observed
> Karlsruhe =
+   c(1.186,1.151,1.322,1.339,1.2,1.402,1.365,1.537,1.559)
> 
> Lehigh = 
+   c(1.061,0.992,1.063,1.062,1.065,1.178,1.037,1.086,1.052)
> 
> # Since the first observations in each vector is from the first girder,
> # the second from the second girder, and so on, the analysis of choice
> # is a paired t-test, where we suppose differences 
> # are N(delta,sigma^2) distributed and test
> # H0: delta = 0
> # aginst
> # H1: delta <> 0,
> # i.e. a two-sided test should be used
> #
> # We analyse 
> dy = Karlsruhe-Lehigh
> 
> plot(
+   c(0,dy),
+   rep(1,1+length(dy)),
+   type = "n",
+   ylab = "",
+   xlab = "Difference between the quotients",
+   axes = FALSE,
+   main = "Figure 1"
+ )
> 
> axis(1)
> grid()
> 
> 
> points(
+   dy,
+   rep(1,length(dy)),
+   pch =16,
+   cex = 1.5,
+   col = "dark red"
+ )
> 
> # As we see NOT A SINGLE observation is below 0, which
> # actually settles the case: there IS actually a difference,.,
> # but let us do the analysis for completeness.
> 
> SD = sd(dy)
> 
> SDerror = SD/sqrt(length(dy))
> MEAN = mean(dy)
> (teststat = MEAN/SDerror)
[1] 6.081939
> 
> # Under H0, the test statistic is t(8) distributed, and the
> # Pvalue is
> #
> (1-pt(teststat,8))*2
[1] 0.0002952955
> 
> # To investigate the normality assumption, I first 
> # make qq-plots separately in the two groups:
> qqnorm(Karlsruhe,main = "Figure 2: QQnorm for Karlruhe")
> 
> qqnorm(Lehigh,main = "Figure 3: QQnorm for Lehigh")
> # This I only did since Montgomery told me to, actually, what 
> # matters is the normality of the differences.
> 
> qqnorm(dy,main = "Figure 4: QQnorm for the difference")
> 
> # f) About the normality assumtion in the
> # paired t-test: First of all, only the difference neads
> # to be normal, so amongst the qqplots, actually only
> # the one in Figure 4 makes sense.. Moreover, the normality 
> # assumtion is not vital anyway.
>