distribution.
A Level Maths Notes: S2 - Testing a Distribution for Goodness of Fit to a Set of 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. We
demonstrate how to test a distribution for goodness of fit to a set
of 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 a
distribution
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 |
|
|
1.35 |
The test statistic is
This is a 90% test. Goodness of fit tests are always 1-tailed. Our
test statistic is less than the critical value. We do not reject
the
distribution
as a possible fit to the data.
