The random variable X has a Poisson distribution with mean \(\mu \). The value of \(\mu \) is known to be either 1 or 2 so the following hypotheses are set up.

\[{{\text{H}}_0}:\mu = 1;{\text{ }}{{\text{H}}_1}:\mu = 2\]

A random sample \({x_1},{\text{ }}{x_2},{\text{ }} \ldots ,{\text{ }}{x_{10}}\) of 10 observations is taken from the distribution of X and the following critical region is defined.

\[\sum\limits_{i = 1}^{10} {{x_i} \geqslant 15} \]

Determine the probability of

(a)     a Type I error;

(b)     a Type II error.

(a)     let \(T = \sum\limits_{i = 1}^{10} {{X_i}} \) so that T is Po(10) under \({{\text{H}}_0}\)     (M1)

\({\text{P(Type I error)}} = {\text{P }}T \geqslant 15|\mu = 1\)     M1A1

\( = 0.0835\)     A2     N3

Note: Candidates who write the first line and only the correct answer award (M1)M0A0A2.

[5 marks]

(b)     let \(T = \sum\limits_{i = 1}^{10} {{X_i}} \) so that T is Po(20) under \({{\text{H}}_1}\)     (M1)

\({\text{P(Type II error)}} = {\text{P }}T \leqslant 14|\mu  = 2\)     M1A1

\( = 0.105\)     A2     N3

Note: Candidates who write the first line and only the correct answer award (M1)M0A0A2.

Note: Award 5 marks to a candidate who confuses Type I and Type II errors and has both answers correct.

[5 marks]

Total [10 marks]

This question caused problems for many candidates and the solutions were often disappointing. Some candidates seemed to be unaware of the meaning of Type I and Type II errors. Others were unable to calculate the probabilities even when they knew what they represented. Candidates who used a normal approximation to obtain the probabilities were not given full credit – there seems little point in using an approximation when the exact value could be found.