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 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.