Statistics: Statistical distributionsEdexcel A-Level Maths: Topic test

1. 1
   A machine selects one ball at random from 20 identical balls numbered 1 to 20. The random variable $X$ is the number on the ball selected.

   (a)

   What is the value of $P(X=7)$?

   [1 mark]

   • A$\dfrac{1}{20}$
   • B$\dfrac{1}{19}$
   • C$\dfrac{7}{20}$
   • D$\dfrac{1}{10}$

   (b)

   What is the value of $P(X\leqslant6)$?

   [1 mark]

   • A$0.25$
   • B$0.35$
   • C$0.3$
   • D$0.05$

   (c)

   Find the probability that $X$ is a multiple of 4 or is greater than 17.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   In a call centre, each call is answered within 20 seconds with probability $0.8$, independently of every other call. The random variable $X$ is the number of calls answered within 20 seconds out of the next 15 calls.

   (a)

   Which distribution is a suitable model for $X$?

   [1 mark]

   • A$B(0.8,15)$
   • B$B(15,0.8)$
   • C$B(15,0.2)$
   • D$N(12,2.4)$

   (b)

   What is the value of $P(X=15)$?

   [1 mark]

   • A$0.8$
   • B$0.6020$
   • C$0.1319$
   • D$0.0352$

   (c)

   Find the probability that at least 13 of the 15 calls are answered within 20 seconds.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The time, $T$ minutes, taken by a runner to complete a 10 km course is modelled by $T\sim N(52,6^2)$.

   (a)

   Find the probability that a runner completes the course in less than 45 minutes.

   [3 marks]

   (b)

   The fastest $5\%$ of runners qualify for a final. Find the slowest time that qualifies.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A bakery sells wholemeal and white loaves. Each customer independently buys a wholemeal loaf with probability $0.48$. On a typical day there are 120 customers, and $X$ is the number who buy a wholemeal loaf.

   (a)

   Use a Normal approximation to $X\sim B(120,0.48)$ to estimate $P(X\geqslant65)$, and state why this approximation is suitable.

   [6 marks]

   (b)

   A group of three customers is chosen at random. Find the probability that at least two of them buy a wholemeal loaf. State two assumptions required for your model and give a reason why one of them may not hold for customers who arrive together.

   [6 marks]

   Total for question 4: 12 marks

5. 5
   The discrete random variable $W$ has probability distribution $P(W=w)=cw^2$ for $w=1,2,3$, where $c$ is a constant.

   (a)

   What is the value of $c$?

   [1 mark]

   • A$\dfrac{1}{6}$
   • B$\dfrac{1}{3}$
   • C$\dfrac{1}{9}$
   • D$\dfrac{1}{14}$

   (b)

   What is the value of $P(W\geqslant2)$?

   [1 mark]

   • A$\dfrac{9}{14}$
   • B$\dfrac{13}{14}$
   • C$\dfrac{5}{14}$
   • D$\dfrac{1}{14}$

   (c)

   Find the probability that $W$ is odd.

   [2 marks]

   Total for question 5: 4 marks

6. 6
   A driving instructor enters 8 students for their theory test. Each student independently passes at the first attempt with probability $0.7$. The random variable $X$ is the number of the 8 students who pass at the first attempt, and $X\sim B(8,0.7)$.

   (a)

   What is the value of $P(X=5)$?

   [1 mark]

   • A$0.168$
   • B$0.00454$
   • C$0.254$
   • D$0.296$

   (b)

   Which expression gives the probability that at least 6 of the students pass at the first attempt?

   [1 mark]

   • A$1-P(X\leqslant5)$
   • B$1-P(X\leqslant6)$
   • C$P(X\leqslant6)$
   • D$1-P(X\leqslant4)$

   (c)

   Find the probability that at least 6 of the 8 students pass at the first attempt.

   [2 marks]

   Total for question 6: 4 marks

7. 7
   The length, $L$ cm, of adult brown trout in a lake is modelled by $L\sim N(\mu,\sigma^2)$. A survey finds that $10\%$ of the trout are shorter than 18 cm and $15\%$ are longer than 30 cm.

   (a)

   Show that the information leads to the equations $18-\mu=-1.2816\sigma$ and $30-\mu=1.0364\sigma$.

   [3 marks]

   (b)

   Solve the equations to find $\mu$ and $\sigma$.

   [4 marks]

   Total for question 7: 7 marks

8. 8
   A weather researcher suggests that the number of rainy days, $R$, in a 30-day June at a coastal town can be modelled by $R\sim B(30,0.3)$.

   (a)

   Using this model, find $P(R=9)$ and $P(R\geqslant12)$, and state two assumptions that the model makes about rainy days.

   [6 marks]

   (b)

   The researcher uses a Normal approximation to estimate $P(R\geqslant12)$. Calculate this estimate, compare it with your exact value from part (a), and comment on whether the approximation and the binomial model are appropriate for rainy days.

   [6 marks]

   Total for question 8: 12 marks

End of questions