18.443, February 6, 2008  The R environment; gravitational constant

          In Athena, you can get into the statistical computing environment
          R by typing "add r"  and then "R"  (without quotes).
          I entered the data on 8 observations of the Newtonian gravitational
          constant from 1987 through 1998 by this command:

          x <- c(6.6670,6.6500,6.6656,6.6685,6.7154,6.6729,6.6740,6.6873)

          "c" can be interpreted as "concatenate." It takes the numbers to
          be components of a vector, and "<-" is the assignment symbol,
          defining the left side to equal the right, like "=" in many
          computer languages. I created a further data vector y to include 
          also the 1981 observation with the command
 
          y <- c(6.1,x)

          which gives a vector with 9 components.

          There is a command in R,   sink("filename") which causes output
          to be written to a file called filename rather than displayed on
          the screen. When you're finished creating the file you then
          type    sink()   to return to displaying output on the screen.
          There is a test of whether data could have reasonably come from
          a normal distribution (we haven't defined the notion of hypothesis
          test yet, but we will later in the course). The test is called
          the Shapiro-Wilk test. I created a file by starting writing to it, 
          then giving the following sequence of commands:

          x
          shapiro.test(x)
          y
          shapiro.test(y)

          Then I stopped writing to the file with   sink() and got out
          of R by  q()  [quit]. Here is the file. R has treated the
          vector x as a 1 by 8 matrix, and y as a 1 by 9 matrix,
          so the [1] denotes first row in each case.

          ---------

          [1] 6.6670 6.6500 6.6656 6.6685 6.7154 6.6729 6.6740 6.6873

	     Shapiro-Wilk normality test

          data:  x 
          W = 0.88, p-value = 0.1881

          [1] 6.1000 6.6670 6.6500 6.6656 6.6685 6.7154 6.6729 6.6740 6.6873

	     Shapiro-Wilk normality test

          data:  y 
          W = 0.4782, p-value = 3.538e-06
 
          - - - - - - - - - -

          The quantity W ("Shapiro-Wilk test statistic," I guess W is Wilk's
          initial) is between 0 and 1. Values closer 
          to 1 indicate a better fit to a normal distribution (which could 
          have any mean and variance). As the number n of observations gets
          larger, W needs to be closer to 1 to be consistent with normality.  
          The p-value is the probability, if the data are i.i.d. normal, of
          observing a W as far from 1 as is observed, for the given number
          of data points (8 or 9 in this case). You can see that for
          x, the p-value is about 0.19, quite substantial, so we wouldn't
          reject the hypothesis that the data in x are normally distributed.
          However, for y, the p-value is less than 4 times 10^{-6}, very 
          small. So for y, we reject the hypothesis of normality. The 
          components of x are grouped rather closely together,
          whereas the observation 6.1 is far from them, it's what's called
          an outlier. In a normal distribution, outliers shouldn't occur.
          The observation 6.1 could have come from a normal distribution
          with the same mean as the others, but a larger variance.