8 The Department of Health 'eat five a day' advice recommends that people should eat at least five portions of fruit and vegetables per day. In a particular school, \(20 \%\) of pupils eat at least five a day.
- 15 children are selected at random.
(A) Find the probability that exactly 3 of them eat at least five a day.
(B) Find the probability that at least 3 of them eat at least five a day.
(C) Find the expected number who eat at least five a day.
A programme is introduced to encourage children to eat more portions of fruit and vegetables per day. At the end of this programme, the diets of a random sample of 15 children are analysed. A hypothesis test is carried out to examine whether the proportion of children in the school who eat at least five a day has increased. - (A) Write down suitable null and alternative hypotheses for the test.
(B) Give a reason for your choice of the alternative hypothesis. - Find the critical region for the test at the \(10 \%\) significance level, showing all of your calculations. Hence complete the test, given that 7 of the 15 children eat at least five a day.