An ecologist tracks the daily migration distance of a species of Arctic tern during its seasonal journeys. During the northward spring migration, the daily distance covered, D D\,D km, can be modelled by a normal distribution D∼N(450,402)D \sim \text{N}(450, 40^2)D∼N(450,402).
Using standardisation and showing your working, find the probability that, on a randomly selected day in spring,
(i) the distance covered is more than 394 km,
(ii) the distance covered is 450 km, correct to the nearest 60 km.
During the southward winter migration, the daily distance W W\,W km is modelled by a normal distribution W∼N(μ,σ2)W \sim \text{N}(\mu, \sigma^2)W∼N(μ,σ2).
Given that P(W>410)=0.0668\text{P}(W > 410) = 0.0668P(W>410)=0.0668 and P(W<330)=0.1587\text{P}(W < 330) = 0.1587P(W<330)=0.1587,
(i) find two equations in terms of μ \mu\,μ and σ\sigmaσ,
(ii) hence, showing your working, find the value of μ \mu\,μ and the value of σ\sigmaσ.
390 exam-style questions on OCR (MEI) A Level Maths 2.4 Probability Distributions, covering 2.4.1 Recognise binomial situations, 2.4.2 Probability of success p, 2.4.3 Calculate binomial probabilities, 2.4.4 Mean of the binomial distribution, 2.4.5 Expected frequencies for binomial, 2.4.6 Probability functions and discrete random variables, 2.4.7 Numerical probabilities for a simple distribution, 2.4.8 Normal distribution as a model (A-level only), 2.4.9 Shape of the Normal curve (A-level only), 2.4.10 Linear transformation and standardising (A-level only), 2.4.11 Symmetry and inflection of Normal curve (A-level only), 2.4.12 Calculate probabilities from a Normal distribution (A-level only), 2.4.13 Model with probability distributions, and 2.4 Probability Distributions. Each one has a worked solution and a mark scheme showing where the marks go.