## Testing a Distribution for Goodness of Fits to Data

Given a set of data, we may suspect intuitively that it may be modelled by a particular model for intuitive reasons, because inspection of the data seems to imply it, or because a particular distribution , if it did fit the data, would be very convenient. Wedemonstrate how to test a distribution for goodness of fit to a setof data using thedistribution.

Suppose we have a list of data. The data represents the number of goals scored in a football tournament is illustrated in the table.

It is proposed to fit adistribution to this table. Conduct a hypothesis test.at the 10% level.

We complete the table below, using the fact that for a Poisson distribution with 100 observations and

We group together the last two columns since the last column has frequency less than 5.

 Number of Goals, k 0 1 2 3 4 5 More than 5 5 or More Observed, O(k) 47 20 15 8 5 5 0 5 Expected, E(k) 30.1 36,1 21.7 8.7 2.6 0.6 3.5 4.1 O(k)-E(k) -16.9 16.1 6.7 0.7 -2.4 -4.4 3.5 -0.9 $(O(k)-E(k))^2$ 285.61 259.21 44.89 0.49 5.76 0.81 9.5 7.2 2.1 0.1 2.21 2.0
$\sum \frac{(O(k)-E(k))^2}{E(k){=23.1$

There are 5 degrees of freedom. Referring to the
$\chi^2$
tables gives a critical balue of 16.75. The Poisson is not a good model.