Homework Problem 19, Due Friday April 2.
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Consider the statistical problem of testing the 
goodness of fit of multinomial data with n=100 
trials and 4 categories 1:4, to a set of cell 
probabilities .1, .2, .3, .4.  Let Obs denote 
the set of 4 observed cell counts.  We know how 
to do a chi-square test of the correctness of 
this set of cell probabilities in (at least) 
two different ways:

(1) calculating sum((Obs-Exp)^2/Exp)  with 
Exp = 100*c(.1,.2,.3,.4), and

(2) calculating the chi-square statistic 
sum((Obs-Exp)^2/Exp)  testing that the cell 
probabilities form an arithmetic progression 
(a, a+b, a+2b, a+3b), with Exp derived from an 
estimated value of a,  obtained by numerical 
maximization of the log-likelihood over a, 
using the fact b=(1-4a)/6.

In (1), the degrees of freedom are 3, and 
in (2) the degrees of freedom are 2. 

Your assignment is to verify through a simulation 
that chi-square distributions with these numbers 
of degrees of freedom closely approximate the 
distributions of the respective test statistics.

Notes. (i) It would be nice to use a large number 
(eg 10000) simulation iterations to do this. 
For this reason, in the interests of moderate 
execution time, you will likely want to parallelize 
the numerical maximizations in situation (2).
(ii) Your results should include an assessment 
of the sampling error in your simulated distribution.
(iii) If you want to check for VERY close 
approximation by the chi-square distributions, you 
should probably re-do your simulations with n=100
sample-size for the Multinomial replaced by n=1000.

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In a little while, we will re-do (parts of) the 
same simulation in SAS.
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