The completion times for three distinct phases of a luxury residential construction project—foundation preparation (FFF), structural framing (GGG), and interior finishing (HHH)—are assumed to follow independent normal distributions:
F∼N(40,4.52) F \sim \text{N}(40, 4.5^2) F∼N(40,4.52) G∼N(60,7.22) G \sim \text{N}(60, 7.2^2) G∼N(60,7.22) H∼N(35,6.42) H \sim \text{N}(35, 6.4^2) H∼N(35,6.42)where the times are measured in working days.
Determine the probability that the total time required for these three phases in a randomly selected project exceeds 150 working days.
Find the probability that the structural framing phase takes at least 30 days longer than the interior finishing phase.
Given that P(F+G+H<t)=0.05P(F + G + H < t) = 0.05P(F+G+H<t)=0.05, find the value of ttt.
A construction firm is overseeing 8 independent projects of this type. Using the probability calculated in part (a), find the probability that the total time for the three phases exceeds 150 working days in at least two of these 8 projects.
An analyst suggests that the assumption of independence between F,G, F, G,\,F,G, and H H\,H within a single project is likely flawed. Explain, with a reason, whether this suggests the model used in part (d) is appropriate.
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.