S1: Mathematical models in probability and statisticsEdexcel International A Level Maths: Topic test

1. 1
   A delivery firm models the journey time from its depot to a warehouse as exactly 40 minutes on every day, ignoring traffic.

   (a)

   Which of the following is an assumption made by the model?

   [1 mark]

   • AThe depot and the warehouse are in the same town.
   • BTraffic and weather conditions do not affect the journey time.
   • CThe journey time is never less than 40 minutes.
   • DA different driver makes the journey each day.

   (b)

   The firm records the actual journey time on 30 days. Which is the best next step in the modelling process?

   [1 mark]

   • AAccept the model because it is simple.
   • BDiscard any recorded time longer than 40 minutes.
   • CChange the recorded times to 40 minutes.
   • DCompare the recorded times with the model's prediction and refine the model if they differ.

   (c)

   The journey times, in minutes, on five days were 38, 41, 45, 52 and 39. Calculate the mean journey time and comment on whether the model is reasonable.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A charity raffle sells 200 tickets, numbered 1 to 200. One ticket is drawn to win first prize. The organiser models the draw by assuming that every ticket is equally likely to be drawn.

   (a)

   Amira buys 5 of the tickets. According to the model, what is the probability that Amira wins first prize?

   [1 mark]

   • A$\frac{1}{40}$
   • B$\frac{1}{200}$
   • C$\frac{1}{5}$
   • D$\frac{5}{195}$

   (b)

   Which of the following would make the model unreasonable?

   [1 mark]

   • AEvery ticket that was sold has a number between 1 and 200.
   • BThe tickets are numbered in the order in which they were sold.
   • CThe tickets are put into a drum without being mixed, with the most recently sold tickets on top, and the winner is taken from the top.
   • DThe organiser draws the winning ticket without looking.

   (c)

   Amira buys 5 tickets in each of 20 similar raffles and wins first prize in 3 of them. Use the model to find the expected number of first prizes she would win in 20 raffles, and comment on the model.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A forecaster models the weather in a town by assuming that, on each day, the probability of rain is 0.2, independently of the weather on any other day.

   (a)

   Using the model, find the probability that it rains on exactly one of the next three days.

   [3 marks]

   (b)

   The forecaster refines the model. On the first day the probability of rain is 0.2. After a rainy day the probability of rain the next day is 0.5, and after a dry day it is 0.1. Using the refined model, find the probability that it rains on exactly one of the first two days, and compare it with the value given by the original model.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   An inspector models the quality of items in a batch of 20 by assuming that each item is defective with probability 0.05, independently of the other items.

   (a)

   (i) State two assumptions made by the model. [2]
   (ii) Find the probability that a batch contains no defective items. [2]
   (iii) In 50 batches, 28 contained no defective items. Compare this with the number predicted by the model. [2]

   [6 marks]

   (b)

   (i) Find the probability that none of the items in three successive batches is defective. [2]
   (ii) Find the probability that a batch contains at least one defective item, and the expected number of such batches among 50. [2]
   (iii) All 50 batches were produced by one machine, which sometimes develops a fault that makes several items in a batch defective together. Explain how this affects the assumption of independence and its effect on the proportion of defect-free batches. [2]

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A sports club models the number of members attending each training session as exactly 25, which was the mean attendance over the first four sessions.

   (a)

   Which of the following is a limitation of the model?

   [1 mark]

   • AIt uses a mean of 25.
   • BIt uses data from the first four sessions only.
   • CIt ignores the variation in attendance from one session to another.
   • DIt cannot be compared with real attendances.

   (b)

   Which term describes the statement 'attendance is the same at every session'?

   [1 mark]

   • AAn assumption of the model.
   • BAn observation.
   • CAn outlier.
   • DA refinement.

   (c)

   The attendances at the next six sessions were 31, 18, 25, 27, 22 and 29. Calculate the mean attendance and comment on the model.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A bakery models the flavour chosen by a customer buying one muffin by the probabilities 0.4 for chocolate, 0.3 for blueberry, 0.2 for lemon and 0.1 for plain.

   (a)

   According to the model, what is the probability that a customer does not choose chocolate?

   [1 mark]

   • A$0.4$
   • B$0.9$
   • C$0.2$
   • D$0.6$

   (b)

   Which check shows that the four probabilities form a valid probability model?

   [1 mark]

   • AEach probability is less than 0.5.
   • BThe probabilities add up to 1.
   • CThe largest probability belongs to the most popular flavour.
   • DAll four probabilities are different.

   (c)

   In a sample of 150 customers, 48 chose chocolate. Use the model to find the expected number choosing chocolate and comment on the result.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A taxi firm models the fare, $F$ pounds, for a journey of $d$ miles by $F=3+2.5d$. The model is intended for journeys of up to 20 miles.

   (a)

   Use the model to find the fare for a journey of 8 miles, and the distance of a journey with a fare of £28.

   [3 marks]

   (b)

   A passenger is charged £68 for a 25-mile journey. Find the fare predicted by the model, explain why this prediction should be treated with caution, and find the percentage by which the model underestimates the charge, as a percentage of the actual charge, to 3 significant figures.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A garden centre models the height, $h$ cm, of a young plant $t$ weeks after planting by $h=5+3t$. The heights of one plant were measured as 11.2 cm after 2 weeks, 17.1 cm after 4 weeks, 26.5 cm after 8 weeks and 33.8 cm after 12 weeks.

   (a)

   (i) Find the heights predicted by the model after 8 weeks and after 12 weeks. [2]
   (ii) By how much does the model overestimate the height at each of these times? [2]
   (iii) Explain why the model is unlikely to be suitable for large values of $t$, and suggest a refinement. [2]

   [6 marks]

   (b)

   (i) Use the model to find when the plant is predicted to reach a height of 50 cm. [1]
   (ii) Find the mean rate of growth, in cm per week, between 8 and 12 weeks, and compare it with the rate in the model. [2]
   (iii) A second model assumes the plant continues to grow at the rate found in part (ii) after 12 weeks. Find its predicted height after 16 weeks. [2]
   (iv) Comment on the reliability of this prediction. [1]

   [6 marks]

   Total for question 8: 12 marks

End of questions