Normal approximation to the binomial and PoissonEdexcel International A Level Further Maths: Subtopic test

1. 1
   The random variable $X$ has distribution $\mathrm{B}(80,0.4)$, and a normal approximation is to be used.

   (a)

   Which normal distribution approximates $X$?

   [1 mark]

   • A$\mathrm{N}(32,32)$
   • B$\mathrm{N}(32,4.38)$
   • C$\mathrm{N}(32,19.2)$
   • D$\mathrm{N}(32,12.8)$

   (b)

   Let $Y\sim\mathrm{N}(32,19.2)$. With a continuity correction, $\mathrm{P}(X\le30)$ is approximated by

   [1 mark]

   • A$\mathrm{P}(Y<29.5)$
   • B$\mathrm{P}(Y<30)$
   • C$\mathrm{P}(Y<31)$
   • D$\mathrm{P}(Y<30.5)$

   (c)

   Hence calculate an approximation to $\mathrm{P}(X\le30)$.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The number of emails $X$ received by a help desk in one hour has distribution $\mathrm{Po}(25)$, and a normal approximation is to be used.

   (a)

   Which normal distribution approximates $X$?

   [1 mark]

   • A$\mathrm{N}(25,25)$
   • B$\mathrm{N}(25,5)$
   • C$\mathrm{N}(5,25)$
   • D$\mathrm{N}(25,625)$

   (b)

   Let $Y\sim\mathrm{N}(25,25)$. With a continuity correction, $\mathrm{P}(X<20)$ is approximated by

   [1 mark]

   • A$\mathrm{P}(Y<20.5)$
   • B$\mathrm{P}(Y<19.5)$
   • C$\mathrm{P}(Y<19)$
   • D$\mathrm{P}(Y<20)$

   (c)

   Hence calculate an approximation to $\mathrm{P}(X\ge30)$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   A component is defective with probability $0.45$, independently of other components. In a random sample of $150$ components, $X$ is the number that are defective.

   (a)

   Using a suitable normal approximation, find $\mathrm{P}(X=70)$.

   [3 marks]

   (b)

   Find an approximation to $\mathrm{P}(60\le X<75)$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   An online shop receives orders at random, and the number of orders $X$ in one day is modelled by $\mathrm{Po}(36)$. Orders on different days are independent. A calculator may be used.

   (a)

   (i) Write down a suitable normal approximation for $X$ and give a reason why it is appropriate.
   (ii) Hence find $\mathrm{P}(30<X\le42)$.

   [6 marks]

   (b)

   The shop starts each day with $k$ items in stock, and each order uses one item.
   (i) Use a normal approximation to find the smallest $k$ such that $\mathrm{P}(X\le k)\ge0.95$.
   (ii) The exact value is $\mathrm{P}(X\le46)=0.956$ to 3 significant figures. Use your normal approximation to find $\mathrm{P}(X\le46)$ and comment on its accuracy.

   [6 marks]

   Total for question 4: 12 marks

End of questions