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{\it Mike Keane -- The $\chi^2$ test for goodness of fit}
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This is not the level of work I was hoping for on the final 
project.  You have not researched this topic very thoroughly, and
what you wrote contains some pretty bad misunderstandings.
The paper itself is quite a bit under the required length.  You
have also not followed any standard method for citing your sources
and documenting where the ideas came from.   

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First, to say how the $\chi^2$ goodness of fit test works in general, 
the idea of ``goodness of fit'' needs to be formulated
in terms of a specific {\it probability model} for how the 
data would be generated.  So the general description of goodness
of fit as considering ``how well the data fits what it should be''
is vague and somewhat inaccurate.  But you do something like the correct approach
in describing how the Mendelian inheritance table would generate
the $9:3:3:1$ ratios for the frequencies of the round/wrinkled and 
yellow/green pea traits depending on the traits of the parent plants.  
So this is not the most serious problem.  (But note that the model
is really the hypothesis here and the question is whether the model
fits that data.)

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Second, the example computation is good up until the final 
interpretation of what the test is telling you.  (By the way, did you
generate this computation yourself, or did you take it from one of 
your sources?   There's no penalty for using results like this as
long as you say where they came from.  But this should be made clear
to the reader!)  The numbers generated here are supposed to be uniformly 
distributed on  the interval 0 - 100, so the bin sizes correspond
to the $9:3:3:1$ ratios correctly.  So this computation is simulating
the kind of data that Mendel {\it should have reported} (see below).  
The computation
of the $\chi^2$ statistic is correct.  But I think you misunderstood 
how you should interpret the results.  

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The null hypothesis for this goodness of fit test is that the 
$9:3:3:1$ ratios {\it do describe the data} reasonably well.  
The alternative is that they {\it do not}.  
This is actually a bit different from the tests we discussed in 
class and you needed to appreciate that to understand what was
going on here.   The larger the $\chi^2$ statistic is, the more evidence
you have to say that the $9:3:3:1$ ratios do not fit.  
But with 3 d.f., the value $\chi^2 = 3.509$ would give a $p$-value 
between $0.5$ and $0.1$.  (Your table gives the probability of 
observing a certain value of $\chi^2$, or something larger,  
if the null hypothesis is true -- that is, if the $9:3:3:1$ model
does describe how the data was generated.)  In other words, there 
{\it is not enough evidence 
to say that the $9:3:3:1$ model does not fit the data} (and of 
course there shouldn't be, because that is how the data
{\it was} generated(!))  We do not reject the null hypothesis in this case.  

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Third, and more seriously (given the length of what you did submit), 
you have not included any of the more detailed
descriptions of where the $\chi^2$ distributions come from, how the
$\chi^2$ tables would be generated, or how 
R.A. Fisher used the $\chi^2$ distributions to analyze Mendel's reported
data.  Recall that the project topic description asked for all of that.  
That would have been interesting, 
because the thing that Fisher found was that Mendel's data was, in a sense,
``too good to be true.''  The agreement with 
the predicted $9:3:3:1$ ratios was {\it so close} that $\chi^2$ came out
very small.   If you look at the $\chi^2$ table, you can see that
equivalently the $p$-value for the test was extremely large, so the 
null hypothesis looked like a sure bet.  
The conclusion that Fisher drew (and that has been debated ever since)
was that Mendel had actually {\it ``fudged''} or {\it ``cherry-picked''} his
data to fit the pattern he thought should be true. Of course we now know
that Mendel was right.  But
if he did that, then he was at the very least not following what we think of 
now as sound scientific methods(!)

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{\it Paper:} 15/20
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{\it Presentation:} 0/5
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{\it Bibliography:} 5/5
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{\it Total:} 20/30  ($67\%$)
\bye  

