6 The number of accidents per month, \(X\), at a factory has a Poisson distribution. In the past the mean has been 1.1 accidents per month. Some new machinery is introduced and the management wish to test whether the mean has increased. They note the number of accidents in a randomly chosen month and carry out a hypothesis test at the 1\% significance level.

1. Show that the critical region for the test is \(X \geqslant 5\). Given that the number of accidents is 6 , carry out the test.
   Later they carry out a similar test, also at the \(1 \%\) significance level.
2. Explain the meaning of a Type I error in this context and state the probability of a Type I error.
3. Given that the mean is now 7.0 , find the probability of a Type II error.
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