7 The number of accidents per year on a certain road has the distribution \(\operatorname { Po } ( \lambda )\). In the past the value of \(\lambda\) was 3.3 . Recently, a new speed limit was imposed and the council wishes to test whether the value of \(\lambda\) has decreased. The council notes the total number, \(X\), of accidents during two randomly chosen years after the speed limit was introduced and it carries out a test at the \(5 \%\) significance level.
- Calculate the probability of a Type I error.
- Given that \(X = 2\), carry out the test.
\includegraphics[max width=\textwidth, alt={}, center]{4f215475-30fd-47fb-aa77-0b53e339f50c-10_2718_35_107_2012}
\includegraphics[max width=\textwidth, alt={}, center]{4f215475-30fd-47fb-aa77-0b53e339f50c-11_2716_29_107_22} - The council decides to carry out another similar test at the \(5 \%\) significance level using the same hypotheses and two different randomly chosen years.
Given that the true value of \(\lambda\) is 0.6 , calculate the probability of a Type II error.
- Using \(\lambda = 0.6\) and a suitable approximating distribution, find the probability that there will be more than 10 accidents in 30 years.
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