4.9 Normal distributionIB Maths: Applications and Interpretation HL: Subtopic test

1. 1
   The mass of rice in a packet is normally distributed with mean $500$ g and standard deviation $8$ g.

   (a)

   Use the $68$–$95$–$99.7$ rule to find the percentage of packets with a mass between $484$ g and $516$ g.

   [1 mark]

   • A$95\%$
   • B$68\%$
   • C$99.7\%$
   • D$47.5\%$

   (b)

   Approximately what percentage of packets have a mass greater than $508$ g?

   [1 mark]

   • A$5\%$
   • B$16\%$
   • C$32\%$
   • D$34\%$

   (c)

   Use your GDC to find the probability that a packet has a mass less than $490$ g.

   [2 marks]

   Total for question 1: 4 marks

2. 2
   The time, $T$ minutes, taken by a commuter train to complete its journey is normally distributed with mean $42$ and standard deviation $6$.

   (a)

   Use your GDC to find the probability that a journey takes more than $50$ minutes.

   [1 mark]

   • A$0.909$
   • B$0.182$
   • C$0.0912$
   • D$0.159$

   (b)

   The longest $10\%$ of journeys take more than $k$ minutes. Find $k$.

   [1 mark]

   • A$34.3$
   • B$48.0$
   • C$54.0$
   • D$49.7$

   (c)

   Use your GDC to find the probability that a journey takes between $40$ and $45$ minutes.

   [2 marks]

   Total for question 2: 4 marks

3. 3
   The mass, $M$ grams, of eggs from a farm is normally distributed with mean $62$ and standard deviation $3$. The eggs are sold in boxes of $240$.

   (a)

   Use your GDC to find the probability that an egg has a mass greater than $67$ g. Hence find the expected number of eggs in a box that are heavier than $67$ g.

   [3 marks]

   (b)

   (i) The heaviest $10\%$ of eggs are graded extra large. Find the least mass of an extra large egg.
   (ii) The lightest $5\%$ of eggs are graded small. Find the greatest mass of a small egg.

   [4 marks]

   Total for question 3: 7 marks

4. 4
   The lifetime, $L$ hours, of a Brand A battery is normally distributed with mean $120$ and standard deviation $15$. The lifetime of a Brand B battery is normally distributed with the same mean and standard deviation $25$. Use your GDC where appropriate.

   (a)

   For Brand A:
   (i) find the probability that a battery lasts less than $96$ hours;
   (ii) write down, with a reason, the approximate probability that a battery lasts between $105$ and $135$ hours;
   (iii) the $2\%$ of batteries with the shortest lifetimes are replaced under guarantee. Find the lifetime below which a battery is replaced.

   [6 marks]

   (b)

   (i) For each brand, find the probability that a battery lasts more than $150$ hours.
   (ii) Explain why the probability is greater for Brand B even though the means are equal.
   (iii) Find, using the $95\%$ rule, the interval centred on the mean that contains approximately $95\%$ of Brand B lifetimes.

   [6 marks]

   Total for question 4: 12 marks

End of questions