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.

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