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.