The Normal distributionEdexcel International A Level Maths: Subtopic test

1. 1
   The mass of an egg, $X$ grams, is Normally distributed with mean 60 and standard deviation 4.

   (a)

   Find the value of $Z=\frac{X-\mu}{\sigma}$ when $X=66$.

   [1 mark]

   • A$6$
   • B$1.5$
   • C$0.375$
   • D$-1.5$

   (b)

   Find $\mathrm{P}(X<66)$.

   [1 mark]

   • A$0.0668$
   • B$0.8664$
   • C$0.4332$
   • D$0.9332$

   (c)

   Find the probability that an egg has a mass between 54 g and 66 g.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The time, $T$ minutes, taken by a bus to complete its route is Normally distributed with $T\sim\mathrm{N}(42,\,9)$.

   (a)

   Write down the standard deviation of $T$.

   [1 mark]

   • A$3$ minutes
   • B$9$ minutes
   • C$81$ minutes
   • D$4.5$ minutes

   (b)

   Find $\mathrm{P}(T>45)$.

   [1 mark]

   • A$0.8413$
   • B$0.1587$
   • C$0.3413$
   • D$0.6306$

   (c)

   Find the value of $t$ such that $\mathrm{P}(T<t)=0.9772$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The heights of adult women in a population are Normally distributed with mean 164 cm and standard deviation 6 cm.

   (a)

   Find the probability that a randomly chosen woman has a height between 158 cm and 173 cm.

   [3 marks]

   (b)

   The tallest 5% of women are classed as tall. Find, to the nearest centimetre, the least height of a woman who is classed as tall.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A machine fills bags of sugar. The mass of sugar in a bag, in grams, is Normally distributed.

   (a)

   Given that 10% of bags have mass less than 995 g and 5% of bags have mass more than 1010 g, find the mean and the standard deviation of the mass of a bag.

   [6 marks]

   (b)

   A second machine fills bags with masses that are Normally distributed with mean 1000 g and standard deviation 5 g.
   (i) A bag is acceptable if its mass is within 10 g of 1000 g. Find the probability that a bag is acceptable.
   (ii) Find the probability that a bag has mass less than 992.5 g.
   (iii) Find the value of $c$ such that 99% of bags have mass less than $c$.

   [6 marks]

   Total for question 4: 12 marks

End of questions