- (i) (a) State the conditions under which the Poisson distribution may be used as an approximation to the binomial distribution.
A factory produces tyres for bicycles and \(0.25 \%\) of the tyres produced are defective.
A company orders 3000 tyres from the factory.
(b) Find, using a Poisson approximation, the probability that there are more than 7 defective tyres in the company’s order.
(ii) At the company \(40 \%\) of employees are known to cycle to work. A random sample of 150 employees is taken. The random variable \(C\) represents the number of employees in the sample who cycle to work.
(a) Describe a suitable sampling frame that can be used to take this sample.
(b) Explain what you understand by the sampling distribution of \(C\)
Louis uses a normal approximation to calculate the probability that at most \(\alpha\) employees in the sample cycle to work. He forgets to use a continuity correction and obtains the incorrect probability 0.0668
Find, showing all stages of your working,
(c) the value of \(\alpha\)
(d) the correct probability.