3 It is known that \(25 \%\) of students in a particular city are smokers. A random sample of 20 of the students is selected.
- (A) Find the probability that there are exactly 4 smokers in the sample.
(B) Find the probability that there are at least 3 but no more than 6 smokers in the sample
(C) Write down the expected number of smokers in the sample.
A new health education programme is introduced. This programme aims to reduce the percentage of students in this city who are smokers. After the programme has been running for a year, it is decided to carry out a hypothesis test to assess the effectiveness of the programme. A random sample of 20 students is selected. - (A) Write down suitable null and alternative hypotheses for the test.
(B) Explain why the alternative hypothesis has the form that it does - Find the critical region for the test at the \(5 \%\) level, showing all of your calculations.
- In fact there are 3 smokers in the sample. Complete the test, stating your conclusion clearly.