1. A machine fills packets with sweets and \(\frac { 1 } { 7 }\) of the packets also contain a prize.

The packets of sweets are placed in boxes before being delivered to shops. There are 40 packets of sweets in each box. The random variable \(T\) represents the number of packets of sweets that contain a prize in each box.

1. State a condition needed for \(T\) to be modelled by \(\mathrm { B } \left( 40 , \frac { 1 } { 7 } \right)\) A box is selected at random.
2. Using \(T \sim \mathrm {~B} \left( 40 , \frac { 1 } { 7 } \right)\) find
   1. the probability that the box has exactly 6 packets containing a prize,
   2. the probability that the box has fewer than 3 packets containing a prize. Kamil's sweet shop buys 5 boxes of these sweets.
3. Find the probability that exactly 2 of these 5 boxes have fewer than 3 packets containing a prize. Kamil claims that the proportion of packets containing a prize is less than \(\frac { 1 } { 7 }\)
   A random sample of 110 packets is taken and 9 packets contain a prize.
4. Use a suitable test to assess Kamil's claim. You should
   • state your hypotheses clearly
   • use a \(5 \%\) level of significance