2. The discrete random variable \(X \sim \mathrm {~B} ( 30,0.28 )\)
- Find \(\mathrm { P } ( 5 \leq X < 12 )\).
Past records from a large supermarket show that \(25 \%\) of people who buy eggs, buy organic eggs. On one particular day a random sample of 40 people is taken from those that had bought eggs and 16 people are found to have bought organic eggs.
- Test, at the \(1 \%\) significance level, whether or not the proportion \(p\) of people who bought organic eggs that day had increased. State your hypotheses clearly.
- State the conclusion you would have reached if a \(5 \%\) significance level had been used for this test.
\section*{(Total for Question 2 is 8 marks)}