We now consider a hypothesis test about whether such a Poisson model underlies a set of counts,
Test statistic
The chi-squared statistic can be used as a test statistic.
\[ X^2 \;\;=\;\; \sum_{i=1}^k {\frac{\left(O_i - E_i\right)^2}{E_i}} \]since
P-value and conclusion
The p-value is the probability that the test statistic, \(X^2\), is as large as was recorded from the actual data, \(x^2\),
\[ \text{p-value} \;\;=\;\; P(X^2 \ge x^2) \]when H0 is true. This can be found from the upper tail of the \(\ChiSqrDistn(k \text{ df})\) distribution.
Example
The following table describes the number of heart attacks in a city in ten weeks.
Week | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
---|---|---|---|---|---|---|---|---|---|---|
Count | 6 | 11 | 13 | 10 | 21 | 8 | 16 | 6 | 9 | 19 |
Test whether the heart attacks occurred at random with a rate of \(\lambda = 10\) per week.
(Solved in full version)