An automatic machine is used to fill bottles of water. The amount delivered, \(Y\) ml , may be assumed to be normally distributed with mean $\mu$ ml and standard deviation $8$ ml . Initially, the machine is adjusted so that the value of $\mu$ is $500$. In order to check that the value of $\mu$ remains equal to $500$, a random sample of 10 bottles is selected at regular intervals, and the mean amount of water, $\overline y$ , in these bottles is calculated. The following hypotheses are set up.

${{\rm{H}}_0}:\mu  = 500$ ; ${{\rm{H}}_1}:\mu  \ne 500$

The critical region is defined to be $\left( {\overline y  < 495} \right) \cup \left( {\overline y  > 505} \right)$ .

(i)     Find the significance level of this procedure.

(ii)     Some time later, the actual value of $\mu$ is $503$. Find the probability of a Type II error.

(i)     Under ${{\rm{H}}_0}$ , the distribution of ${\overline y }$ is N(500, 6.4) .     (A1)

Significance level $= {\rm{P}}\overline y  < 495$ or $> 505|{{\rm{H}}_0}$     M2

$= 2 \times 0.02405$     (A1)

$= 0.0481$     A1 N5

Note: Using tables, answer is $0.0478$.

(ii)     The distribution of $\overline y$ is now N($503$, $6.4$) .     (A1)

P(Type ΙΙ error) $= {\rm{P}}(495 < \overline y  < 505)$     (M1)

$= 0.785$     A1 N3

Note: Using tables, answer is $0.784$.

[8 marks]