19 Ellie, a statistics student, read a newspaper article that stated that 20 per cent of students eat at least five portions of fruit and vegetables every day.
Ellie suggests that the number of people who eat at least five portions of fruit and vegetables every day, in a sample of size \(n\), can be modelled by the binomial distribution \(\mathrm { B } ( n , 0.20 )\).
19
- There are 10 students in Ellie's statistics class.
Using the distributional model suggested by Ellie, find the probability that, of the students in her class:
19 - two or fewer eat at least five portions of fruit and vegetables every day;
[0pt]
[1 mark]
19
- (ii) at least one but fewer than four eat at least five portions of fruit and vegetables every day;
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[2 marks]
19 - Ellie's teacher, Declan, believes that more than 20 per cent of students eat at least five portions of fruit and vegetables every day. Declan asks the 25 students in his other statistics classes and 8 of them say that they eat at least five portions of fruit and vegetables every day.
19
- Name the sampling method used by Declan.
19
- (ii) Describe one weakness of this sampling method.
19 - (iii) Assuming that these 25 students may be considered to be a random sample, carry out a hypothesis test at the \(5 \%\) significance level to investigate whether Declan's belief is supported by this evidence.
[0pt]
[6 marks]