1. 1
   The masses of apples from an orchard are normally distributed with mean 150 g and standard deviation 12 g. Let $X$ be the mass, in grams, of an apple chosen at random, so that $X\sim N(150,\,12^{2})$. A calculator may be used in part (c) only.

   (a)

   Estimate the percentage of apples with a mass between 126 g and 174 g.

   [1 mark]

   • A68%
   • B47.5%
   • C99.7%
   • D95%

   (b)

   Estimate the percentage of apples with a mass greater than 162 g.

   [1 mark]

   • A34%
   • B32%
   • C16%
   • D84%

   (c)

   Find the probability that a randomly chosen apple has a mass between 140 g and 165 g.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The time, $T$ minutes, that a commuter's bus journey takes is normally distributed with mean 42 minutes and standard deviation 5 minutes. A calculator may be used in this question.

   (a)

   Which one of the following statements is true?

   [1 mark]

   • A$P(T<35)=P(T>49)$
   • B$P(T<35)=P(T<49)$
   • C$P(T<42)=0.68$
   • D$P(T>47)=0.34$

   (b)

   Find $P(T<35)$, correct to 3 significant figures.

   [1 mark]

   • A$0.919$
   • B$0.0808$
   • C$0.838$
   • D$0.0404$

   (c)

   Only 10% of the commuter's journeys take longer than $k$ minutes. Find $k$.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The lifetime of a type of battery is normally distributed with mean 620 hours and standard deviation 45 hours. A calculator may be used in this question.

   (a)

   A shop receives a delivery of 400 of these batteries. Find the expected number of batteries in the delivery that last less than 550 hours.

   [3 marks]

   (b)

   The manufacturer will replace, free of charge, any battery that lasts less than $g$ hours, where $g$ is a whole number. It wants to replace no more than 2% of batteries. Find the greatest possible value of $g$, justifying your answer.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   A machine fills bags of flour that are labelled 1000 g. The mass of flour in a bag is normally distributed with mean 1012 g and standard deviation 8 g. A bag is underweight if it contains less than 1000 g. The masses of different bags are independent. A calculator may be used in this question.

   (a)

   (i) Find the probability that a bag chosen at random is underweight.
   (ii) A box contains 12 bags. Find the probability that the box contains at least one underweight bag.
   (iii) Find the mass that is exceeded by 99% of the bags.

   [6 marks]

   (b)

   (i) A bag is chosen at random and is found not to be underweight. Find the probability that it contains more than 1020 g.
   (ii) Find the interquartile range of the masses of flour in the bags.

   [6 marks]

   Total for question 4: 12 marks

End of questions