In the experiment on pea breeding
Mendel's theory predicts the probabilities of
occurrence associated with the types of progeny, say
``round yellow'', ``wrinkled yellow'', ``round green'', and
Here we want to test whether the data from
is consistent with his theory--goodness of fit
The model probabilities
are specified (usually in the column Probability
categories or ``cells.''
Out of the total size n
each observation is classified into one of the k
and the expected cell frequencies
are calculated from the model probabilities
The observed cell frequencies
gives the total size
of cell frequencies.
Then the goodness of fit to the model
can be assessed by comparing the
observed cell frequencies with
the expected cell frequencies.
Here the statement of null hypothesis becomes ``the model is valid.''
The discrepancy between the data and the model
can be measured by the Pearson's chi-square statistic
Under the null hypothesis
(that is, assuming that the model probabilities are correct),
the distribution of Pearson's chi-square
is approximated by the chi-square distribution
degrees of freedom.
we can reject the null hypothesis
if you observe that
casting doubt on the validity of the model.
Or equivalently, by computing the -value
with a random variable having the chi-square distribution
with degrees of freedom,
we can find that the null hypothesis is rejected if
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