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]