6. The Headteacher of a school claims that \(30 \%\) of parents do not support a new curriculum. In a survey of 20 randomly selected parents, the number, \(X\), who do not support the new curriculum is recorded.
Assuming that the Headteacher's claim is correct, find
- the probability that \(X = 5\)
- the mean and the standard deviation of \(X\)
The Director of Studies believes that the proportion of parents who do not support the new curriculum is greater than \(30 \%\). Given that in the survey of 20 parents 8 do not support the new curriculum,
- test, at the \(5 \%\) level of significance, the Director of Studies' belief. State your hypotheses clearly.
The teachers believe that the sample in the original survey was biased and claim that only \(25 \%\) of the parents are in support of the new curriculum. A second random sample, of size \(2 n\), is taken and exactly half of this sample supports the new curriculum.
A test is carried out at a 10\% level of significance of the teachers' belief using this sample of size \(2 n\)
Using the hypotheses \(\mathrm { H } _ { 0 } : p = 0.25\) and \(\mathrm { H } _ { 1 } : p > 0.25\)
- find the minimum value of \(n\) for which the outcome of the test is that the teachers' belief is rejected.