6 A biscuit manufacturer claims that, on average, 1 in 3 packets of biscuits contain a prize offer. Gerry suspects that the proportion of packets containing the prize offer is less than 1 in 3 . In order to test the manufacturer's claim, he buys 20 randomly selected packets. He finds that exactly 2 of these packets contain the prize offer.

1. Carry out the test at the \(10 \%\) significance level.
2. Maria also suspects that the proportion of packets containing the prize offer is less than 1 in 3 . She also carries out a significance test at the \(10 \%\) level using 20 randomly selected packets. She will reject the manufacturer's claim if she finds that there are 3 or fewer packets containing the prize offer. Find the probability of a Type II error in Maria's test if the proportion of packets containing the prize offer is actually 1 in 7 .
3. Explain what is meant by a Type II error in this context.