The normal distributionEdexcel A-Level Maths: Subtopic test

1. 1
   The height, $H$ cm, of a plant of a certain variety is modelled by $H\sim N(42,3^2)$.

   (a)

   Find $P(H>45)$.

   [1 mark]

   • A$0.841$
   • B$0.159$
   • C$0.341$
   • D$0.023$

   (b)

   At which values of $H$ does the graph of the probability density function have points of inflection?

   [1 mark]

   • A$H=39$ and $H=45$
   • B$H=36$ and $H=48$
   • C$H=-3$ and $H=3$
   • D$H=42$ only

   (c)

   Find the probability that a plant chosen at random has a height between 38 cm and 47 cm.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   Scores on a reasoning test are modelled by $X\sim N(100,15^2)$.

   (a)

   Find the standardised value $z$ for a score of 130.

   [1 mark]

   • A$30$
   • B$0.5$
   • C$0.13$
   • D$2$

   (b)

   Find $P(X>130)$.

   [1 mark]

   • A$0.9772$
   • B$0.0456$
   • C$0.0228$
   • D$0.1587$

   (c)

   Find the score that is exceeded by 10% of people.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The lifetime, $L$ hours, of a type of battery is modelled by $L\sim N(\mu,\sigma^2)$. It is found that $P(L<200)=0.1$ and $P(L>260)=0.2$.

   (a)

   Show that $\mu-1.2816\sigma=200$ and $\mu+0.8416\sigma=260$.

   [3 marks]

   (b)

   Hence find the values of $\mu$ and $\sigma$.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A machine fills bottles of drink. The volume, $V$ ml, of drink in a bottle is modelled by $V\sim N(330,2.5^2)$. The label on each bottle states 330 ml.

   (a)

   (i) Find $P(V<327)$.
   (ii) Find $P(328<V<334)$.
   (iii) The mean is adjusted to 332 ml with the standard deviation unchanged. Find the new probability that a bottle contains less than 327 ml.

   [6 marks]

   (b)

   The manufacturer requires fewer than 1% of bottles to contain less than 325 ml.
   (i) Show that the original machine, with $V\sim N(330,2.5^2)$, does not meet this requirement.
   (ii) With the mean kept at 330 ml, find the largest standard deviation for which the requirement is met.

   [6 marks]

   Total for question 4: 12 marks

End of questions