We now formally test the hypotheses
The following test statistic is used,
\[ X^2 \;\;=\;\; 2\log(R) \;\;=\;\; 2\left(\ell(\mathcal{M}_B) - \ell(\mathcal{M}_S)\right) \]\(X^2\) has (approximately) a standard distribution when H0 holds, and is likely to be largest if \(\mathcal{M}_S\) is not correct.
Distribution of test statistic
If the data do come from \(L(\mathcal{M}_S)\), and \(L(\mathcal{M}_B)\) has \(k\) more parameters than \(L(\mathcal{M}_S)\),
\[ X^2 \;\;=\;\; 2\left( \ell(\mathcal{M}_B) - \ell(\mathcal{M}_S)\right) \;\;\underset{\text{approx}}{\sim} \;\; \ChiSqrDistn(k \text{ df}) \]Likelihood ratio test
Question
The following table describes the number of defective items from a production line in each of 20 days.
1 2 |
3 4 |
2 3 |
2 5 |
5 2 |
4 3 |
5 1 |
2 4 |
0 2 |
2 6 |
Assuming that the data are a random sample from a \(\PoissonDistn(\lambda)\) distribution, use a likelihood ratio test for whether the rate of defects was \(\lambda = 2\) per week.
(Solved in full version)