The random variable \(X\) has a Poisson distribution with parameter \(\lambda\).
Prove, from first principles, that \(\mathrm { E } ( X ) = \lambda\).
Given that \(\mathrm { E } \left( X ^ { 2 } - X \right) = \lambda ^ { 2 }\), deduce that \(\operatorname { Var } ( X ) = \lambda\).
The number of faults in a 100-metre ball of nylon string may be modelled by a Poisson distribution with parameter \(\lambda\).
An analysis of one ball of string, selected at random, showed 15 faults.
Using an exact test, investigate the claim that \(\lambda > 10\). Use the \(5 \%\) level of significance.
A subsequent analysis of a random sample of 20 balls of string showed a total of 241 faults.
(A) Using an approximate test, re-investigate the claim that \(\lambda > 10\). Use the \(5 \%\) level of significance.
(B) Determine the critical value of the total number of faults for the test in part (b)(ii)(A).
(C) Given that, in fact, \(\lambda = 12\), estimate the probability of a Type II error for a test of the claim that \(\lambda > 10\) based upon a random sample of 20 balls of string and using the \(5 \%\) level of significance. [0pt]
[4 marks]
\includegraphics[max width=\textwidth, alt={}, center]{d5852425-9340-4aae-82da-e3bf6772a0de-22_2490_1728_219_141}
\includegraphics[max width=\textwidth, alt={}, center]{d5852425-9340-4aae-82da-e3bf6772a0de-23_2490_1719_217_150}
\includegraphics[max width=\textwidth, alt={}, center]{d5852425-9340-4aae-82da-e3bf6772a0de-24_2489_1728_221_141}