6 A factory makes chocolates of different types. The proportion of milk chocolates made on any day is denoted by \(p\). It is desired to test the null hypothesis \(\mathrm { H } _ { 0 } : p = 0.8\) against the alternative hypothesis \(\mathrm { H } _ { 1 } : p < 0.8\). The test consists of choosing a random sample of 25 chocolates. \(\mathrm { H } _ { 0 }\) is rejected if the number of milk chocolates is \(k\) or fewer. The test is carried out at a significance level as close to \(5 \%\) as possible.

1. Use tables to find the value of \(k\), giving the values of any relevant probabilities.
2. The test is carried out 20 times, and each time the value of \(p\) is 0.8 . Each of the tests is independent of all the others. State the expected number of times that the test will result in rejection of the null hypothesis.
3. The test is carried out once. If in fact the value of \(p\) is 0.6 , find the probability of rejecting \(\mathrm { H } _ { 0 }\).
4. The test is carried out twice. Each time the value of \(p\) is equally likely to be 0.8 or 0.6 . Find the probability that exactly one of the two tests results in rejection of the null hypothesis.