Statistical modellingEdexcel International A Level Maths: Subtopic test

1. 1
   A game designer models the result $X$ of a single spin of a spinner that has five sectors, numbered 1 to 5.

   (a)

   The model uses $P(X=x)=\frac15$ for $x=1,2,3,4,5$. Which assumption does this model make?

   [1 mark]

   • ALarger numbers are more likely, with $P(X=x)=\frac{x}{15}$
   • BEach spin depends on the result of the previous spin
   • CThe five sectors have different sizes
   • DEach sector is equally likely to be landed on

   (b)

   Using the model $P(X=x)=\frac15$, find the expected number of times that a 3 occurs in 200 spins.

   [1 mark]

   • A$20$
   • B$40$
   • C$60$
   • D$120$

   (c)

   In 200 spins a 3 occurs 41 times. Comment on whether this result supports the model.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   A bus company states that its 08:15 bus is late with probability 0.15 on any day. A commuter uses this statement as a model for the number of late buses over 20 weekdays.

   (a)

   Which of these is an assumption of the model?

   [1 mark]

   • AThe probability that the bus is late is the same every day, and whether it is late on one day does not affect other days
   • BThe bus is late on exactly 3 of the 20 days
   • CThe bus is never late on two days in a row
   • DThe bus is late more often on Mondays

   (b)

   Which observation gives the strongest evidence that the model is unsuitable?

   [1 mark]

   • AThe bus was late on 4 of the 20 days
   • BThe bus was on time on the first day
   • CWhen the bus was late on one day it was usually late on the next day as well
   • DThe bus was late on 2 of the 20 days

   (c)

   Find the number of late buses that the model predicts over the 20 days. In fact the bus was late on 9 of the 20 days. Comment on the suitability of the model.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A teacher models the score $X$ on one roll of a die by $P(X=x)=\frac16$ for $x=1,2,3,4,5,6$. A student rolls the die 60 times. The scores 1, 2, 3, 4, 5 and 6 occur 7, 12, 9, 11, 8 and 13 times respectively.

   (a)

   (i) State the expected frequency of each score under the model.
   (ii) Find the relative frequency of the score 6 from the student's data.

   [3 marks]

   (b)

   Comment on how well the data fit the model, and suggest how the student could improve the investigation.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   80 students each toss a coin four times and record the number of heads, $X$. The numbers of students who obtained 0, 1, 2, 3 and 4 heads were 3, 15, 33, 23 and 6 respectively. A teacher models $X$ by assuming that all 16 possible sequences of heads and tails are equally likely.

   (a)

   (i) State two conditions on the coin and the tosses that justify the model.
   (ii) Show that, under the model, $P(X=2)=\frac38$.
   (iii) Hence find the expected number of students who obtain exactly two heads.

   [6 marks]

   (b)

   Evaluate how well the model fits the students' results, and suggest how the investigation could be improved.

   [6 marks]

   Total for question 4: 12 marks

End of questions