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24 exam-style questions with model answers, plus 32 quick multi-choice questions — every question on the site for this standard, grouped by the 8 pages of notes they come from.
Write a full answer before you reveal the model one — that comparison is where the marks come from. Every block links back to the notes that teach it.
A distribution has P(X=1) = 0.22, P(X=2) = 0.31, P(X=3) = k and P(X=4) = 0.19. Find k, then find P(X ≥ 3).
A café sells a set breakfast in three sizes. Records show 45% of customers choose small ($12), 35% medium ($16) and 20% large ($21). Let X be the amount a randomly chosen customer spends. Write the distribution, find P(X > 12), and explain whether X is discrete or continuous.
A charity raffle sells 500 tickets at $5 each. There is one prize of $600, three prizes of $100 and ten prizes of $20. A buyer says: 'With 14 prizes among 500 tickets, my chance of winning something is nearly 3%, and the prizes total $1 100, so the raffle is roughly fair.' Discuss the buyer's reasoning.
A random variable has P(X=0) = 0.3, P(X=1) = 0.5, P(X=2) = 0.2. Find E(X) and the standard deviation.
A tour operator charges $85 per passenger and has fixed costs of $900 per trip. Passenger numbers per trip have mean 18 and standard deviation 4. Find the mean and standard deviation of profit, and comment on the risk.
A courier company's parcels have mass with mean 2.4 kg and standard deviation 0.9 kg. A van carries 60 randomly selected parcels. A manager calculates the total load's standard deviation as 60 × 0.9 = 54 kg and concludes the load is far too unpredictable to plan around. Discuss.
A basketball player makes 65% of free throws. In 12 attempts, find the probability she makes exactly 9, and the mean number made.
A seed company claims 92% germination. A gardener plants 40 seeds and only 32 germinate. Calculate the probability of 32 or fewer germinating if the claim is true, and comment on the claim.
An airline knows that 8% of passengers with bookings fail to turn up. It sells 265 tickets for a 250-seat flight. Find the probability that more passengers turn up than there are seats, and advise the airline on its overbooking policy.
Potholes occur at an average rate of 2.5 per kilometre on a rural road. Find the probability of exactly 3 potholes in one kilometre, and the standard deviation.
A bakery sells an average of 18 pies per hour. Find the probability that fewer than 5 are sold in a 15-minute period, and explain why the Poisson model may not hold across a whole day.
A hospital emergency department is designing its overnight staffing. Arrivals average 4.5 per hour between midnight and 6am. A manager argues that since 6 hours × 4.5 = 27 arrivals are expected and one doctor can see 5 patients per hour (30 over the shift), one doctor is sufficient. Evaluate this reasoning.
A variable X is rectangular between 20 and 50. Find the density height, P(X < 30) and the mean.
A machine cuts lengths of timber that are modelled as rectangular between 2.38 m and 2.42 m. Lengths outside 2.39 m to 2.41 m are rejected. Find the rejection rate and the standard deviation, and comment on whether a rectangular model is likely to be realistic.
A power company models the time of a random equipment fault along a 40 km transmission line as rectangular between 0 and 40 km from the substation. It uses this to argue that a repair depot should be placed at the 20 km midpoint to minimise average travel distance. Evaluate the model and the conclusion.
A triangular distribution has a = 5, c = 8, b = 20. Find the peak height and the mean.
Daily rainfall at a site on a rainy day is modelled as triangular with a minimum of 0 mm, a maximum of 60 mm and a mode of 10 mm. Find the probability of more than 25 mm, and comment on whether the model is sensible.
A construction firm estimates that a project phase will take a minimum of 20 days, a maximum of 50 days, and most likely 24 days, and models the duration as triangular. The project manager promises the client completion in 30 days, saying '30 is well above the most likely 24, so we have plenty of margin.' Assess this reasoning and advise the manager.
Heights of a group are normally distributed with mean 170 cm and standard deviation 8 cm. Find the z-score for 182 cm and the probability a randomly chosen person is taller than 182 cm.
A coffee machine dispenses volumes that are normally distributed with standard deviation 4 mL. The owner wants no more than 2% of cups to contain less than 240 mL. Find the mean the machine should be set to, and comment on the cost of this setting.
A school reports that its Year 13 students' scores on an internal test have mean 62 and standard deviation 15, and models them as normal. The deputy principal uses this to predict that about 2.3% of the cohort scored above 92, and that about 2.3% scored below 32. The actual counts in a cohort of 180 were 11 students above 92 and 2 students below 32. Assess the model.
For each, name the distribution: (i) the number of sixes in 20 rolls of a die; (ii) the number of emails arriving in an hour; (iii) the heights of adult women.
A supermarket models the number of customers joining a checkout queue in a 5-minute period as Poisson with λ = 6. Over 200 five-minute periods it observes a mean of 6.1 and a variance of 6.4. Assess the model, and explain how this differs from having a large sample.
A council models the number of days per year on which a river exceeds a flood warning level as Poisson with λ = 2.5, based on the last 40 years averaging 2.5 such days annually. It uses the model to conclude that a year with 8 or more warning days has probability 0.0042, or about once in 240 years, and designs its emergency response capacity accordingly. Evaluate this.