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 A Level Maths 2.4 Statistical Distributions, covering 2.4.1 Discrete probability distributions, 2.4.2 Binomial distribution as a model, 2.4.3 Calculating binomial probabilities, 2.4.4 Mean and variance of the binomial (A-level only), 2.4.5 Normal distribution as a model (A-level only), 2.4.6 Probabilities using the normal distribution (A-level only), 2.4.7 Links to histograms, mean and standard deviation (A-level only), and 2.4.8 Selecting an appropriate distribution (A-level only). Each one has a worked solution and a mark scheme showing where the marks go.