Batches of specialized coffee beans are monitored for a rare quality trait called 'extra-large' size after roasting. Historical data indicates that the probability of a bean being extra-large is 0.015.
Write down a suitable model for the distribution of the number of extra-large beans in a batch of n n\,n beans.
State one assumption required for this model to be valid in this context.
Using a suitable approximation, find the probability that exactly 7 out of 400 beans in a random batch are identified as being extra-large.
Explain why the approximation used in part (c) is appropriate.
A head barista claims that 70% of regular customers prefer oat milk over dairy milk in their lattes.
In a random sample of 150 regular customers, it is found that 92 prefer oat milk.
Using a suitable approximation, test the barista's claim at the 5% significance level. State your hypotheses clearly.
