Comments on Assignment 6 
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1.(a) Pearson's X^2 = 9.55 which leads to p-value = 0.049.

  (b) Fisher's exact test leads to p-value = 0.064 (the probability of 
      all tables which have the same margins as the observed table but 
      are "as or more extreme" relative to Ho).

  (c) The calculation of a p-value for Pearson's X^2 statistics is based 
      on an asymptotic approximation (basically the CLT -- always!) but 
      the p-value for Fisher's exact test is based on exact calculations 
      (always!), so in that sense Fisher's is ALWAYS more reliable.  In 
      this example, they lead to quite similar p-values (0.05 vs 0.06).
      In terms of summarizing the strength of evidence the data provide 
      against Ho, these are essentially the same.  Yet note that someone
      who rigidly (and blindly!) follows an accept/reject approach to 
      data analysis would come to different conclusions based on the two
      tests in this example.

Note: For this example, all the expected counts (under Ho) are > 5, 
      so not too surprising that the p-values agree fairly well. If this is
      NOT the case, rely more on Fisher's. 


2.(a) Trivial

  (b) Trivial
  
  (c) c=0.001 yields the median (the deviations of essentially all the data
      points from the location estimate are Winsorized to c); c=10 yields 
      the mean (essentially). Look at your results again!!

  (d) The median (or Huber with c=0.001) is most resistant; the mean 
      (or Huber-M with c=10) is the least resistant.

  (e) If the population is normally distributed, the mean is the most 
      efficient (it is the MLE) and the median is the least efficient.
     
  (f) In the sense described above, the Huber's estimate attempts to 
      compromise between resistance and efficiency. Smaller values of 
      c correspond to more resistance and larger values of c correspond 
      to more efficiency.  Default value of c=1.345 is specified to 
      achieve high levels of efficiency in the normal case.