Further Statistics 2: Combinations of random variablesEdexcel A-Level Further Maths: Topic test

1. 1
   A commuter's walk to the station takes $W$ minutes, where $W\sim\mathrm{N}(12,2^2)$, and the train journey takes $T$ minutes, where $T\sim\mathrm{N}(35,3^2)$. The two times are independent.

   (a)

   What is the distribution of the total time $W+T$?

   [1 mark]

   • A$\mathrm{N}(47,5)$
   • B$\mathrm{N}(47,13)$
   • C$\mathrm{N}(47,25)$
   • D$\mathrm{N}(23,13)$

   (b)

   What is the distribution of $T-W$?

   [1 mark]

   • A$\mathrm{N}(23,5)$
   • B$\mathrm{N}(23,-5)$
   • C$\mathrm{N}(47,13)$
   • D$\mathrm{N}(23,13)$

   (c)

   Find the probability that the commuter's walk and train journey together take more than $50$ minutes.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The volume of juice in a carton, $J$ ml, is modelled by $J\sim\mathrm{N}(500,5^2)$. Three cartons are chosen at random, with volumes $J_1$, $J_2$ and $J_3$, which are independent.

   (a)

   What is the distribution of $J_1+J_2+J_3$?

   [1 mark]

   • A$\mathrm{N}(1500,75)$
   • B$\mathrm{N}(1500,225)$
   • C$\mathrm{N}(1500,15)$
   • D$\mathrm{N}(500,75)$

   (b)

   What is the variance of $3J$?

   [1 mark]

   • A$75$
   • B$15$
   • C$225$
   • D$45$

   (c)

   Explain why $J_1+J_2+J_3$ and $3J$ have the same mean but different variances, in the context of the cartons.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   In a swimming relay, four swimmers each swim one leg. Their times in seconds are independent, with $A\sim\mathrm{N}(62.4,1.5^2)$, $B\sim\mathrm{N}(63.1,1.8^2)$, $C\sim\mathrm{N}(61.8,1.2^2)$ and $D\sim\mathrm{N}(64.0,2.0^2)$. The club record for the relay is $248$ s.

   (a)

   Find the distribution of the total relay time $R=A+B+C+D$.

   [3 marks]

   (b)

   Find the probability that the team beats the club record. State one assumption you have made.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A factory makes shafts and bearings. The diameter of a shaft, $S$ mm, is modelled by $S\sim\mathrm{N}(19.98,0.02^2)$, and the diameter of the hole in a bearing, $H$ mm, by $H\sim\mathrm{N}(20.05,0.03^2)$. A shaft fits a bearing if $H>S$. Shafts and bearings are chosen at random and independently.

   (a)

   Find the probability that a randomly chosen shaft fits a randomly chosen bearing.

   [6 marks]

   (b)

   Two bearings, $H_1$ and $H_2$, are fitted on one assembly with two shafts, $S_1$ and $S_2$, so that the total clearance is $T=H_1+H_2-S_1-S_2$. Find the probability that the total clearance exceeds $0.10$ mm, and state one assumption you have made.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   A university module's final mark is $F=0.4C+0.6E$, where the coursework mark $C\sim\mathrm{N}(62,5^2)$ and the exam mark $E\sim\mathrm{N}(55,12^2)$ are independent.

   (a)

   What is the mean of $F$?

   [1 mark]

   • A$58.5$
   • B$117$
   • C$57.8$
   • D$24.8$

   (b)

   What is the variance of $F$?

   [1 mark]

   • A$55.84$
   • B$7.47$
   • C$169$
   • D$9.2$

   (c)

   Find the probability that a student's final mark is greater than $65$.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   At the start of a sprint, an athlete's reaction time is $R_1$ seconds on one start and $R_2$ seconds on a second start, where $R_1$ and $R_2$ are independent and each is distributed $\mathrm{N}(0.18,0.02^2)$.

   (a)

   What is the distribution of $R_1-R_2$?

   [1 mark]

   • A$\mathrm{N}(0,0)$
   • B$\mathrm{N}(0,0.0004)$
   • C$\mathrm{N}(0.36,0.0008)$
   • D$\mathrm{N}(0,0.0008)$

   (b)

   What is the probability that $R_1-R_2>0.04$?

   [1 mark]

   • A$0.0228$
   • B$0.0786$
   • C$0.1587$
   • D$0.9214$

   (c)

   Find the probability that the two reaction times differ by more than $0.04$ seconds.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   A loaded pallet consists of an empty pallet of mass $P$ kg, where $P\sim\mathrm{N}(20,1.5^2)$, carrying $10$ crates, each of mass $C$ kg, where $C\sim\mathrm{N}(12,0.5^2)$. The masses of the pallet and of all the crates are independent.

   (a)

   Find the distribution of the total mass $M$ of a loaded pallet.

   [3 marks]

   (b)

   A lorry carries three independent loaded pallets and is overloaded if their total mass exceeds $430$ kg. Find the probability that the lorry is overloaded.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A cereal manufacturer's packets from one machine have mass $X$ g, where $X\sim\mathrm{N}(400,6^2)$. Packets from a second machine have mass $Y$ g, where $Y\sim\mathrm{N}(410,9^2)$. All packets are independent.

   (a)

   Five packets are chosen at random from the first machine. Find the probability that their mean mass is less than $397$ g.

   [6 marks]

   (b)

   A packet is chosen at random from each machine. Find the probability that the packet from the second machine is more than $15$ g heavier than the packet from the first. A retailer claims that every packet from the second machine is heavier than every packet from the first. Use a further calculation to comment on this claim.

   [6 marks]

   Total for question 8: 12 marks

End of questions