In a peat bog, Common Spotted-orchids occur at a mean rate of 4.5 per $$\(^{2}\)$$ (a) Give an assumption, not already stated, that is required for the number of Common Spotted-orchids per $$\(\mathrm{m}^{2}\)$$ of the peat bog to follow a Poisson distribution. (1) Given that the number of Common Spotted-orchids in $$\(1 \mathrm{~m}^{2}\)$$ of the peat bog can be modelled by a Poisson distribution, (b) find the probability that in a randomly selected $$\(1 \mathrm{~m}^{2}\)$$ of the peat bog (i) there are exactly 6 Common Spotted-orchids, (ii) there are fewer than 10 but more than 4 Common Spotted-orchids. (4) Juan believes that by introducing a new management scheme the number of Common Spotted-orchids in the peat bog will increase. After three years under the new management scheme, a randomly selected $$\(2 \mathrm{~m}^{2}\)$$ of the peat bog contains 11 Common Spotted-orchids. (c) Using a 5\% significance level assess Juan's belief. State your hypotheses clearly. (5) Assuming that in the peat bog, Common Spotted-orchids still occur at a mean rate of 4.5 per $$\(\mathrm{m}^{2}\)$$ (d) use a normal approximation to find the probability that in a randomly selected $$\(20 \mathrm{~m}^{2}\)$$ of the peat bog there are fewer than 70 Common Spotted-orchids. (3) Following a period of dry weather, the probability that there are fewer than $$\(70 \mathrm{Common}\)$$ Spotted-orchids in a randomly selected $$\(20 \mathrm{~m}^{2}\)$$ of the peat bog is 0.012 A random sample of 200 non-overlapping $$\(20 \mathrm{~m}^{2}\)$$ areas of the peat bog is taken. (e) Using a suitable approximation, calculate the probability that at most 1 of these areas contains fewer than 70 Common Spotted-orchids. (3)

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