## Testing a Distribution for Goodness of Fits to Data

Given a set of data, we may suspect intuitively that it may bemodelled by a particular model for intuitive reasons, becauseinspection of the data seems to imply it, or because a particulardistribution , 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 ofgoals scored in a football tournament is illustrated in the table.

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

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

We group together the last two columns since the last column hasfrequency 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 | 1.35 |